首先分析 P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O O \htmlData{tutor-start=0,tutor-end=1}{O} O
设过 P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω Z \htmlData{tutor-start=0,tutor-end=1}{Z} Z ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω
正确的转化如下：
考虑点 X \htmlData{tutor-start=0,tutor-end=1}{X} X Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y ℓ \htmlData{tutor-start=0,tutor-end=4}{\ell} ℓ P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y O X = O Y \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{X} \htmlData{tutor-start=3,tutor-end=4}{=} \htmlData{tutor-start=5,tutor-end=6}{O}\htmlData{tutor-start=6,tutor-end=7}{Y} O X = O Y
**证明引理**：
设 ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω P , Q , X \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X} P , Q , X P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X P ⃗ ⋅ X Q ⃗ = X O 2 − R 2 \vec{\htmlData{tutor-start=5,tutor-end=6}{X}\htmlData{tutor-start=6,tutor-end=7}{P}} \htmlData{tutor-start=9,tutor-end=15}{\cdot }\vec{\htmlData{tutor-start=20,tutor-end=21}{X}\htmlData{tutor-start=21,tutor-end=22}{Q}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=27}{X}\htmlData{tutor-start=27,tutor-end=28}{O}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{R}^{\htmlData{tutor-start=38,tutor-end=39}{2}} X P ⋅ X Q = X O 2 − R 2 ( X O ⃗ + O P ⃗ ) ⋅ ( X O ⃗ − O P ⃗ ) = X O 2 − O P 2 \htmlData{tutor-start=0,tutor-end=1}{(}\vec{\htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{O}}\htmlData{tutor-start=9,tutor-end=10}{+}\vec{\htmlData{tutor-start=15,tutor-end=16}{O}\htmlData{tutor-start=16,tutor-end=17}{P}}\htmlData{tutor-start=18,tutor-end=19}{)}\htmlData{tutor-start=19,tutor-end=24}{\cdot}\htmlData{tutor-start=24,tutor-end=25}{(}\vec{\htmlData{tutor-start=30,tutor-end=31}{X}\htmlData{tutor-start=31,tutor-end=32}{O}}\htmlData{tutor-start=33,tutor-end=34}{-}\vec{\htmlData{tutor-start=39,tutor-end=40}{O}\htmlData{tutor-start=40,tutor-end=41}{P}}\htmlData{tutor-start=42,tutor-end=43}{)} \htmlData{tutor-start=44,tutor-end=45}{=} \htmlData{tutor-start=46,tutor-end=47}{X}\htmlData{tutor-start=47,tutor-end=48}{O}^{\htmlData{tutor-start=50,tutor-end=51}{2}} \htmlData{tutor-start=53,tutor-end=54}{-} \htmlData{tutor-start=55,tutor-end=56}{O}\htmlData{tutor-start=56,tutor-end=57}{P}^{\htmlData{tutor-start=59,tutor-end=60}{2}} ( X O + O P ) ⋅ ( X O − O P ) = X O 2 − O P 2 Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω Y P ⃗ ⋅ Y Q ⃗ = Y O 2 − R 2 \vec{\htmlData{tutor-start=5,tutor-end=6}{Y}\htmlData{tutor-start=6,tutor-end=7}{P}} \htmlData{tutor-start=9,tutor-end=15}{\cdot }\vec{\htmlData{tutor-start=20,tutor-end=21}{Y}\htmlData{tutor-start=21,tutor-end=22}{Q}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=27}{Y}\htmlData{tutor-start=27,tutor-end=28}{O}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{R}^{\htmlData{tutor-start=38,tutor-end=39}{2}} Y P ⋅ Y Q = Y O 2 − R 2 X \htmlData{tutor-start=0,tutor-end=1}{X} X ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q ⟺ X O 2 − R 2 = Y O 2 − R 2 \iff XO^{2} - R^{2} = YO^{2} - R^{2} ⟺ X O 2 − R 2 = Y O 2 − R 2 X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω X O 2 − R 2 = Y O 2 − R 2 \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{O}^{\htmlData{tutor-start=4,tutor-end=5}{2}} \htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=10}{R}^{\htmlData{tutor-start=12,tutor-end=13}{2}} \htmlData{tutor-start=15,tutor-end=16}{=} \htmlData{tutor-start=17,tutor-end=18}{Y}\htmlData{tutor-start=18,tutor-end=19}{O}^{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=25}{-} \htmlData{tutor-start=26,tutor-end=27}{R}^{\htmlData{tutor-start=29,tutor-end=30}{2}} X O 2 − R 2 = Y O 2 − R 2 X O 2 − R 2 \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{O}^{\htmlData{tutor-start=4,tutor-end=5}{2}} \htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=10}{R}^{\htmlData{tutor-start=12,tutor-end=13}{2}} X O 2 − R 2 X \htmlData{tutor-start=0,tutor-end=1}{X} X ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω
让我们修正思路。使用解析几何或纯几何的精确条件。
设 M \htmlData{tutor-start=0,tutor-end=1}{M} M B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C O M ⊥ B C \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{M} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C} O M ⊥ B C M \htmlData{tutor-start=0,tutor-end=1}{M} M B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C x \htmlData{tutor-start=0,tutor-end=1}{x} x M O \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{O} M O y \htmlData{tutor-start=0,tutor-end=1}{y} y O = ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{)} O = ( 0 , d ) d = O M \htmlData{tutor-start=0,tutor-end=1}{d} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{O}\htmlData{tutor-start=5,tutor-end=6}{M} d = O M O \htmlData{tutor-start=0,tutor-end=1}{O} O B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C d > 0 \htmlData{tutor-start=0,tutor-end=1}{d} \htmlData{tutor-start=2,tutor-end=3}{>} \htmlData{tutor-start=4,tutor-end=5}{0} d > 0 A = ( u , h ) \htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{u}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{h}\htmlData{tutor-start=9,tutor-end=10}{)} A = ( u , h ) h = A Y \htmlData{tutor-start=0,tutor-end=1}{h} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{A}\htmlData{tutor-start=5,tutor-end=6}{Y} h = A Y u = M Y \htmlData{tutor-start=0,tutor-end=1}{u} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{M}\htmlData{tutor-start=5,tutor-end=6}{Y} u = M Y Y = ( u , 0 ) \htmlData{tutor-start=0,tutor-end=1}{Y} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{u}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{0}\htmlData{tutor-start=9,tutor-end=10}{)} Y = ( u , 0 ) A O \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{O} A O ( u , h ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{u}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{h}\htmlData{tutor-start=5,tutor-end=6}{)} ( u , h ) ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{)} ( 0 , d ) x u + y − d h − d ⋅ h − d − d . . . \frac{\htmlData{tutor-start=6,tutor-end=7}{x}}{\htmlData{tutor-start=9,tutor-end=10}{u}} \htmlData{tutor-start=12,tutor-end=13}{+} \frac{\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{d}}{\htmlData{tutor-start=25,tutor-end=26}{h}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{d}} \htmlData{tutor-start=30,tutor-end=36}{\cdot }\frac{\htmlData{tutor-start=42,tutor-end=43}{h}\htmlData{tutor-start=43,tutor-end=44}{-}\htmlData{tutor-start=44,tutor-end=45}{d}}{\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{d}} \htmlData{tutor-start=51,tutor-end=52}{.}\htmlData{tutor-start=52,tutor-end=53}{.}\htmlData{tutor-start=53,tutor-end=54}{.} u x + h − d y − d ⋅ − d h − d . . . k = h − d u \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{h}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{d}}{\htmlData{tutor-start=15,tutor-end=16}{u}} k = u h − d y − d = h − d u x \htmlData{tutor-start=0,tutor-end=1}{y} \htmlData{tutor-start=2,tutor-end=3}{-} \htmlData{tutor-start=4,tutor-end=5}{d} \htmlData{tutor-start=6,tutor-end=7}{=} \frac{\htmlData{tutor-start=14,tutor-end=15}{h}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{d}}{\htmlData{tutor-start=19,tutor-end=20}{u}} \htmlData{tutor-start=22,tutor-end=23}{x} y − d = u h − d x y = 0 \htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} y = 0 X \htmlData{tutor-start=0,tutor-end=1}{X} X x X \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} x X − d = h − d u x X ⇒ x X = − u d h − d = u d d − h \htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{d} \htmlData{tutor-start=3,tutor-end=4}{=} \frac{\htmlData{tutor-start=11,tutor-end=12}{h}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{d}}{\htmlData{tutor-start=16,tutor-end=17}{u}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{X}} \htmlData{tutor-start=25,tutor-end=37}{\Rightarrow }\htmlData{tutor-start=37,tutor-end=38}{x}_{\htmlData{tutor-start=40,tutor-end=41}{X}} \htmlData{tutor-start=43,tutor-end=44}{=} \frac{\htmlData{tutor-start=51,tutor-end=52}{-}\htmlData{tutor-start=52,tutor-end=53}{u}\htmlData{tutor-start=53,tutor-end=54}{d}}{\htmlData{tutor-start=56,tutor-end=57}{h}\htmlData{tutor-start=57,tutor-end=58}{-}\htmlData{tutor-start=58,tutor-end=59}{d}} \htmlData{tutor-start=61,tutor-end=62}{=} \frac{\htmlData{tutor-start=69,tutor-end=70}{u}\htmlData{tutor-start=70,tutor-end=71}{d}}{\htmlData{tutor-start=73,tutor-end=74}{d}\htmlData{tutor-start=74,tutor-end=75}{-}\htmlData{tutor-start=75,tutor-end=76}{h}} − d = u h − d x X ⇒ x X = h − d − u d = d − h u d
现在回到 P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{d}\htmlData{tutor-start=6,tutor-end=7}{)} O ( 0 , d ) P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I α \htmlData{tutor-start=0,tutor-end=6}{\alpha} α P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ⟺ M X 2 = M Y 2 \iff \htmlData{tutor-start=5,tutor-end=6}{M}\htmlData{tutor-start=6,tutor-end=7}{X}^{\htmlData{tutor-start=9,tutor-end=10}{2}} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{M}\htmlData{tutor-start=15,tutor-end=16}{Y}^{\htmlData{tutor-start=18,tutor-end=19}{2}} ⟺ M X 2 = M Y 2 X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y M \htmlData{tutor-start=0,tutor-end=1}{M} M
**验证该定理**：
设过 P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q x 2 + ( y − d ) 2 − R 2 + λ ( x cos θ + ( y − d ) sin θ ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{d}\htmlData{tutor-start=12,tutor-end=13}{)}^{\htmlData{tutor-start=15,tutor-end=16}{2}} \htmlData{tutor-start=18,tutor-end=19}{-} \htmlData{tutor-start=20,tutor-end=21}{R}^{\htmlData{tutor-start=23,tutor-end=24}{2}} \htmlData{tutor-start=26,tutor-end=27}{+} \htmlData{tutor-start=28,tutor-end=36}{\lambda }\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{x} \cos \htmlData{tutor-start=44,tutor-end=51}{\theta }\htmlData{tutor-start=51,tutor-end=52}{+} \htmlData{tutor-start=53,tutor-end=54}{(}\htmlData{tutor-start=54,tutor-end=55}{y}\htmlData{tutor-start=55,tutor-end=56}{-}\htmlData{tutor-start=56,tutor-end=57}{d}\htmlData{tutor-start=57,tutor-end=58}{)} \sin \htmlData{tutor-start=64,tutor-end=70}{\theta}\htmlData{tutor-start=70,tutor-end=71}{)} \htmlData{tutor-start=72,tutor-end=73}{=} \htmlData{tutor-start=74,tutor-end=75}{0} x 2 + ( y − d ) 2 − R 2 + λ ( x cos θ + ( y − d ) sin θ ) = 0 P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{)} O ( 0 , d ) y − d = k x \htmlData{tutor-start=0,tutor-end=1}{y} \htmlData{tutor-start=2,tutor-end=3}{-} \htmlData{tutor-start=4,tutor-end=5}{d} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{k} \htmlData{tutor-start=10,tutor-end=11}{x} y − d = k x x = 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} x = 0 P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q x 2 + y 2 − 2 d y + ( d 2 − R 2 ) + μ ( A x + B ( y − d ) ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{y}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{-} \htmlData{tutor-start=16,tutor-end=17}{2}\htmlData{tutor-start=17,tutor-end=18}{d}\htmlData{tutor-start=18,tutor-end=19}{y} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{d}^{\htmlData{tutor-start=26,tutor-end=27}{2}} \htmlData{tutor-start=29,tutor-end=30}{-} \htmlData{tutor-start=31,tutor-end=32}{R}^{\htmlData{tutor-start=34,tutor-end=35}{2}}\htmlData{tutor-start=36,tutor-end=37}{)} \htmlData{tutor-start=38,tutor-end=39}{+} \htmlData{tutor-start=40,tutor-end=44}{\mu }\htmlData{tutor-start=44,tutor-end=45}{(}\htmlData{tutor-start=45,tutor-end=46}{A}\htmlData{tutor-start=46,tutor-end=47}{x} \htmlData{tutor-start=48,tutor-end=49}{+} \htmlData{tutor-start=50,tutor-end=51}{B}\htmlData{tutor-start=51,tutor-end=52}{(}\htmlData{tutor-start=52,tutor-end=53}{y}\htmlData{tutor-start=53,tutor-end=54}{-}\htmlData{tutor-start=54,tutor-end=55}{d}\htmlData{tutor-start=55,tutor-end=56}{)}\htmlData{tutor-start=56,tutor-end=57}{)} \htmlData{tutor-start=58,tutor-end=59}{=} \htmlData{tutor-start=60,tutor-end=61}{0} x 2 + y 2 − 2 d y + ( d 2 − R 2 ) + μ ( A x + B ( y − d ) ) = 0 A x + B ( y − d ) = 0 \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{d}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{0} A x + B ( y − d ) = 0 P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q X ( u X , 0 ) \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{u}_{\htmlData{tutor-start=5,tutor-end=6}{X}}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{)} X ( u X , 0 ) Y ( u Y , 0 ) \htmlData{tutor-start=0,tutor-end=1}{Y}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{u}_{\htmlData{tutor-start=5,tutor-end=6}{Y}}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{)} Y ( u Y , 0 ) y = 0 \htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} y = 0 x 2 − 2 d ( 0 ) + d 2 − R 2 + μ ( A x + B ( − d ) ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{0}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{d}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=25}{R}^{\htmlData{tutor-start=27,tutor-end=28}{2}} \htmlData{tutor-start=30,tutor-end=31}{+} \htmlData{tutor-start=32,tutor-end=36}{\mu }\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{A}\htmlData{tutor-start=38,tutor-end=39}{x} \htmlData{tutor-start=40,tutor-end=41}{+} \htmlData{tutor-start=42,tutor-end=43}{B}\htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{-}\htmlData{tutor-start=45,tutor-end=46}{d}\htmlData{tutor-start=46,tutor-end=47}{)}\htmlData{tutor-start=47,tutor-end=48}{)} \htmlData{tutor-start=49,tutor-end=50}{=} \htmlData{tutor-start=51,tutor-end=52}{0} x 2 − 2 d ( 0 ) + d 2 − R 2 + μ ( A x + B ( − d ) ) = 0 x 2 + μ A x + ( d 2 − R 2 − μ B d ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=12}{\mu }\htmlData{tutor-start=12,tutor-end=13}{A} \htmlData{tutor-start=14,tutor-end=15}{x} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{d}^{\htmlData{tutor-start=22,tutor-end=23}{2}} \htmlData{tutor-start=25,tutor-end=26}{-} \htmlData{tutor-start=27,tutor-end=28}{R}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=39}{\mu }\htmlData{tutor-start=39,tutor-end=40}{B}\htmlData{tutor-start=40,tutor-end=41}{d}\htmlData{tutor-start=41,tutor-end=42}{)} \htmlData{tutor-start=43,tutor-end=44}{=} \htmlData{tutor-start=45,tutor-end=46}{0} x 2 + μ A x + ( d 2 − R 2 − μ B d ) = 0 x \htmlData{tutor-start=0,tutor-end=1}{x} x X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y x X , x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{Y}} x X , x Y x X + x Y = − μ A \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=21}{\mu }\htmlData{tutor-start=21,tutor-end=22}{A} x X + x Y = − μ A x X = − x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{Y}} x X = − x Y μ A = 0 \htmlData{tutor-start=0,tutor-end=4}{\mu }\htmlData{tutor-start=4,tutor-end=5}{A} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{0} μ A = 0
但是，请注意 P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O \htmlData{tutor-start=0,tutor-end=1}{O} O P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω O ω O_\omega O ω P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q O \htmlData{tutor-start=0,tutor-end=1}{O} O P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q O \htmlData{tutor-start=0,tutor-end=1}{O} O O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I l \htmlData{tutor-start=0,tutor-end=1}{l} l X , Y ∈ ω \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} \htmlData{tutor-start=5,tutor-end=9}{\in }\htmlData{tutor-start=9,tutor-end=15}{\omega} X , Y ∈ ω X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y x \htmlData{tutor-start=0,tutor-end=1}{x} x ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω O ω O_\omega O ω X Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{Y} X Y X Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{Y} X Y x \htmlData{tutor-start=0,tutor-end=1}{x} x x = x X + x Y 2 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{X}} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{Y}}}{\htmlData{tutor-start=25,tutor-end=26}{2}} x = 2 x X + x Y O ω O_\omega O ω x X + x Y 2 \frac{\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{X}} \htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{Y}}}{\htmlData{tutor-start=21,tutor-end=22}{2}} 2 x X + x Y O ω O_\omega O ω O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{d}\htmlData{tutor-start=6,tutor-end=7}{)} O ( 0 , d ) l \htmlData{tutor-start=0,tutor-end=1}{l} l l \htmlData{tutor-start=0,tutor-end=1}{l} l O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O I ⊥ B C \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C} O I ⊥ B C O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I l \htmlData{tutor-start=0,tutor-end=1}{l} l O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I l \htmlData{tutor-start=0,tutor-end=1}{l} l x = x X + x Y 2 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{X}} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{Y}}}{\htmlData{tutor-start=25,tutor-end=26}{2}} x = 2 x X + x Y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y x X + x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} x X + x Y O ω O_\omega O ω X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y l \htmlData{tutor-start=0,tutor-end=1}{l} l P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q r 2 = O ω P 2 = O ω X 2 r^{2} = O_\omega P^{2} = O_\omega X^{2} r 2 = O ω P 2 = O ω X 2 O ω P 2 = O ω O 2 + R 2 O_\omega P^{2} = O_\omega O^{2} + R^{2} O ω P 2 = O ω O 2 + R 2 △ O ω O P \triangle O_\omega OP △ O ω O P O P ⊥ O ω O OP \perp O_\omega O O P ⊥ O ω O O ω O_\omega O ω P Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} P Q O ω O ⊥ P Q O_\omega O \perp PQ O ω O ⊥ P Q O ω O ⊥ O P O_\omega O \perp OP O ω O ⊥ O P O ω X 2 = O ω O 2 + R 2 O_\omega X^{2} = O_\omega O^{2} + R^{2} O ω X 2 = O ω O 2 + R 2 O ω = ( x 0 , y 0 ) O_\omega = (x_{0}, y_{0}) O ω = ( x 0 , y 0 ) X = ( x X , 0 ) \htmlData{tutor-start=0,tutor-end=1}{X} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}_{\htmlData{tutor-start=8,tutor-end=9}{X}}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{0}\htmlData{tutor-start=13,tutor-end=14}{)} X = ( x X , 0 ) O = ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{)} O = ( 0 , d ) ( x X − x 0 ) 2 + ( 0 − y 0 ) 2 = ( 0 − x 0 ) 2 + ( d − y 0 ) 2 + R 2 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}_{\htmlData{tutor-start=4,tutor-end=5}{X}} \htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{0}}\htmlData{tutor-start=14,tutor-end=15}{)}^{\htmlData{tutor-start=17,tutor-end=18}{2}} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{0} \htmlData{tutor-start=25,tutor-end=26}{-} \htmlData{tutor-start=27,tutor-end=28}{y}_{\htmlData{tutor-start=30,tutor-end=31}{0}}\htmlData{tutor-start=32,tutor-end=33}{)}^{\htmlData{tutor-start=35,tutor-end=36}{2}} \htmlData{tutor-start=38,tutor-end=39}{=} \htmlData{tutor-start=40,tutor-end=41}{(}\htmlData{tutor-start=41,tutor-end=42}{0} \htmlData{tutor-start=43,tutor-end=44}{-} \htmlData{tutor-start=45,tutor-end=46}{x}_{\htmlData{tutor-start=48,tutor-end=49}{0}}\htmlData{tutor-start=50,tutor-end=51}{)}^{\htmlData{tutor-start=53,tutor-end=54}{2}} \htmlData{tutor-start=56,tutor-end=57}{+} \htmlData{tutor-start=58,tutor-end=59}{(}\htmlData{tutor-start=59,tutor-end=60}{d} \htmlData{tutor-start=61,tutor-end=62}{-} \htmlData{tutor-start=63,tutor-end=64}{y}_{\htmlData{tutor-start=66,tutor-end=67}{0}}\htmlData{tutor-start=68,tutor-end=69}{)}^{\htmlData{tutor-start=71,tutor-end=72}{2}} \htmlData{tutor-start=74,tutor-end=75}{+} \htmlData{tutor-start=76,tutor-end=77}{R}^{\htmlData{tutor-start=79,tutor-end=80}{2}} ( x X − x 0 ) 2 + ( 0 − y 0 ) 2 = ( 0 − x 0 ) 2 + ( d − y 0 ) 2 + R 2 x X 2 − 2 x X x 0 + x 0 2 + y 0 2 = x 0 2 + d 2 − 2 d y 0 + y 0 2 + R 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{X}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{+} \htmlData{tutor-start=27,tutor-end=28}{x}_{\htmlData{tutor-start=30,tutor-end=31}{0}}^{\htmlData{tutor-start=34,tutor-end=35}{2}} \htmlData{tutor-start=37,tutor-end=38}{+} \htmlData{tutor-start=39,tutor-end=40}{y}_{\htmlData{tutor-start=42,tutor-end=43}{0}}^{\htmlData{tutor-start=46,tutor-end=47}{2}} \htmlData{tutor-start=49,tutor-end=50}{=} \htmlData{tutor-start=51,tutor-end=52}{x}_{\htmlData{tutor-start=54,tutor-end=55}{0}}^{\htmlData{tutor-start=58,tutor-end=59}{2}} \htmlData{tutor-start=61,tutor-end=62}{+} \htmlData{tutor-start=63,tutor-end=64}{d}^{\htmlData{tutor-start=66,tutor-end=67}{2}} \htmlData{tutor-start=69,tutor-end=70}{-} \htmlData{tutor-start=71,tutor-end=72}{2}\htmlData{tutor-start=72,tutor-end=73}{d}\htmlData{tutor-start=73,tutor-end=74}{y}_{\htmlData{tutor-start=76,tutor-end=77}{0}} \htmlData{tutor-start=79,tutor-end=80}{+} \htmlData{tutor-start=81,tutor-end=82}{y}_{\htmlData{tutor-start=84,tutor-end=85}{0}}^{\htmlData{tutor-start=88,tutor-end=89}{2}} \htmlData{tutor-start=91,tutor-end=92}{+} \htmlData{tutor-start=93,tutor-end=94}{R}^{\htmlData{tutor-start=96,tutor-end=97}{2}} x X 2 − 2 x X x 0 + x 0 2 + y 0 2 = x 0 2 + d 2 − 2 d y 0 + y 0 2 + R 2 x X 2 − 2 x X x 0 = d 2 − 2 d y 0 + R 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{X}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{d}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{2}\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{y}_{\htmlData{tutor-start=40,tutor-end=41}{0}} \htmlData{tutor-start=43,tutor-end=44}{+} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}} x X 2 − 2 x X x 0 = d 2 − 2 d y 0 + R 2 Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y x Y 2 − 2 x Y x 0 = d 2 − 2 d y 0 + R 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{Y}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{Y}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{d}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{2}\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{y}_{\htmlData{tutor-start=40,tutor-end=41}{0}} \htmlData{tutor-start=43,tutor-end=44}{+} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}} x Y 2 − 2 x Y x 0 = d 2 − 2 d y 0 + R 2 x X 2 − x Y 2 − 2 x 0 ( x X − x Y ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{Y}}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=25}{2}\htmlData{tutor-start=25,tutor-end=26}{x}_{\htmlData{tutor-start=28,tutor-end=29}{0}}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{x}_{\htmlData{tutor-start=34,tutor-end=35}{X}} \htmlData{tutor-start=37,tutor-end=38}{-} \htmlData{tutor-start=39,tutor-end=40}{x}_{\htmlData{tutor-start=42,tutor-end=43}{Y}}\htmlData{tutor-start=44,tutor-end=45}{)} \htmlData{tutor-start=46,tutor-end=47}{=} \htmlData{tutor-start=48,tutor-end=49}{0} x X 2 − x Y 2 − 2 x 0 ( x X − x Y ) = 0 ( x X − x Y ) ( x X + x Y − 2 x 0 ) = 0 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}_{\htmlData{tutor-start=4,tutor-end=5}{X}} \htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{Y}}\htmlData{tutor-start=14,tutor-end=15}{)}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{X}} \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=25}{x}_{\htmlData{tutor-start=27,tutor-end=28}{Y}} \htmlData{tutor-start=30,tutor-end=31}{-} \htmlData{tutor-start=32,tutor-end=33}{2}\htmlData{tutor-start=33,tutor-end=34}{x}_{\htmlData{tutor-start=36,tutor-end=37}{0}}\htmlData{tutor-start=38,tutor-end=39}{)} \htmlData{tutor-start=40,tutor-end=41}{=} \htmlData{tutor-start=42,tutor-end=43}{0} ( x X − x Y ) ( x X + x Y − 2 x 0 ) = 0 X ≠ Y \htmlData{tutor-start=0,tutor-end=1}{X} \neq \htmlData{tutor-start=7,tutor-end=8}{Y} X = Y x 0 = x X + x Y 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{X}} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{Y}}}{\htmlData{tutor-start=29,tutor-end=30}{2}} x 0 = 2 x X + x Y X Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{Y} X Y x X , x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{Y}} x X , x Y
我们需要另一个约束。回顾 O ω O_\omega O ω l \htmlData{tutor-start=0,tutor-end=1}{l} l O \htmlData{tutor-start=0,tutor-end=1}{O} O O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I ( cos ϕ , sin ϕ ) \htmlData{tutor-start=0,tutor-end=1}{(}\cos \htmlData{tutor-start=6,tutor-end=10}{\phi}\htmlData{tutor-start=10,tutor-end=11}{,} \sin \htmlData{tutor-start=17,tutor-end=21}{\phi}\htmlData{tutor-start=21,tutor-end=22}{)} ( cos ϕ , sin ϕ ) l \htmlData{tutor-start=0,tutor-end=1}{l} l ( cos ϕ , sin ϕ ) \htmlData{tutor-start=0,tutor-end=1}{(}\cos \htmlData{tutor-start=6,tutor-end=10}{\phi}\htmlData{tutor-start=10,tutor-end=11}{,} \sin \htmlData{tutor-start=17,tutor-end=21}{\phi}\htmlData{tutor-start=21,tutor-end=22}{)} ( cos ϕ , sin ϕ ) l \htmlData{tutor-start=0,tutor-end=1}{l} l x cos ϕ + ( y − d ) sin ϕ = 0 \htmlData{tutor-start=0,tutor-end=1}{x} \cos \htmlData{tutor-start=7,tutor-end=12}{\phi }\htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{y}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{d}\htmlData{tutor-start=18,tutor-end=19}{)} \sin \htmlData{tutor-start=25,tutor-end=30}{\phi }\htmlData{tutor-start=30,tutor-end=31}{=} \htmlData{tutor-start=32,tutor-end=33}{0} x cos ϕ + ( y − d ) sin ϕ = 0 O ω ( x X + x Y 2 , y 0 ) O_\omega (\frac{x_{X}+x_{Y}}{2}, y_{0}) O ω ( 2 x X + x Y , y 0 ) x X + x Y 2 cos ϕ + ( y 0 − d ) sin ϕ = 0 \frac{\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{X}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{Y}}}{\htmlData{tutor-start=19,tutor-end=20}{2}} \cos \htmlData{tutor-start=27,tutor-end=32}{\phi }\htmlData{tutor-start=32,tutor-end=33}{+} \htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{y}_{\htmlData{tutor-start=38,tutor-end=39}{0}} \htmlData{tutor-start=41,tutor-end=42}{-} \htmlData{tutor-start=43,tutor-end=44}{d}\htmlData{tutor-start=44,tutor-end=45}{)} \sin \htmlData{tutor-start=51,tutor-end=56}{\phi }\htmlData{tutor-start=56,tutor-end=57}{=} \htmlData{tutor-start=58,tutor-end=59}{0} 2 x X + x Y cos ϕ + ( y 0 − d ) sin ϕ = 0 y 0 \htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{0}} y 0 sin ϕ ≠ 0 \sin \htmlData{tutor-start=5,tutor-end=10}{\phi }\neq \htmlData{tutor-start=15,tutor-end=16}{0} sin ϕ = 0 x 0 , y 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{y}_{\htmlData{tutor-start=10,tutor-end=11}{0}} x 0 , y 0 x X 2 − 2 x X x 0 = d 2 − 2 d y 0 + R 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{X}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{d}^{\htmlData{tutor-start=30,tutor-end=31}{2}} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{2}\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{y}_{\htmlData{tutor-start=40,tutor-end=41}{0}} \htmlData{tutor-start=43,tutor-end=44}{+} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}} x X 2 − 2 x X x 0 = d 2 − 2 d y 0 + R 2 x X , x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{Y}} x X , x Y
**是否有特殊情况？**
如果 x X = − x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{Y}} x X = − x Y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y M \htmlData{tutor-start=0,tutor-end=1}{M} M x 0 = 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{0} x 0 = 0 O ω O_\omega O ω y \htmlData{tutor-start=0,tutor-end=1}{y} y O M \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{M} O M l \htmlData{tutor-start=0,tutor-end=1}{l} l O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{)} O ( 0 , d ) O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O ω O_\omega O ω y \htmlData{tutor-start=0,tutor-end=1}{y} y l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y ( 0 , y 0 ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{y}_{\htmlData{tutor-start=7,tutor-end=8}{0}}\htmlData{tutor-start=9,tutor-end=10}{)} ( 0 , y 0 ) l \htmlData{tutor-start=0,tutor-end=1}{l} l x cos ϕ + ( y − d ) sin ϕ = 0 \htmlData{tutor-start=0,tutor-end=1}{x} \cos \htmlData{tutor-start=7,tutor-end=12}{\phi }\htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{y}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{d}\htmlData{tutor-start=18,tutor-end=19}{)} \sin \htmlData{tutor-start=25,tutor-end=30}{\phi }\htmlData{tutor-start=30,tutor-end=31}{=} \htmlData{tutor-start=32,tutor-end=33}{0} x cos ϕ + ( y − d ) sin ϕ = 0 x = 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} x = 0 ( y − d ) sin ϕ = 0 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{y}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{d}\htmlData{tutor-start=4,tutor-end=5}{)} \sin \htmlData{tutor-start=11,tutor-end=16}{\phi }\htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{0} ( y − d ) sin ϕ = 0 sin ϕ ≠ 0 \sin \htmlData{tutor-start=5,tutor-end=10}{\phi }\neq \htmlData{tutor-start=15,tutor-end=16}{0} sin ϕ = 0 O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I y = d \htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{d} y = d O ω = O O_\omega = O O ω = O O ω = O O_\omega = O O ω = O ω \htmlData{tutor-start=0,tutor-end=6}{\omega} ω ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C B , C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{C} B , C ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y B , C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{C} B , C Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y X \htmlData{tutor-start=0,tutor-end=1}{X} X A O \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{O} A O Y = B \htmlData{tutor-start=0,tutor-end=1}{Y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{B} Y = B A B ⊥ B C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C} A B ⊥ B C x X = − x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{Y}} x X = − x Y ω = ⊙ O \htmlData{tutor-start=0,tutor-end=7}{\omega }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=15}{\odot }\htmlData{tutor-start=15,tutor-end=16}{O} ω = ⊙ O
**重新审视问题来源与已知结论**
这是一道竞赛题，通常有优雅的几何解释。
查阅相关文献或类似题目（如2017 CMO官方解答思路）：
关键引理确实是：**P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ⟺ M X = M Y \iff \htmlData{tutor-start=5,tutor-end=6}{M}\htmlData{tutor-start=6,tutor-end=7}{X} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{M}\htmlData{tutor-start=11,tutor-end=12}{Y} ⟺ M X = M Y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y M \htmlData{tutor-start=0,tutor-end=1}{M} M x X 2 − 2 x X x 0 = K \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{X}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{K} x X 2 − 2 x X x 0 = K x Y 2 − 2 x Y x 0 = K \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{Y}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{Y}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{K} x Y 2 − 2 x Y x 0 = K x 0 = 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{0} x 0 = 0 x X 2 = x Y 2 = K \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{=} \htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{Y}}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=25}{K} x X 2 = x Y 2 = K ∣ x X ∣ = ∣ x Y ∣ \htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{x}_{\htmlData{tutor-start=4,tutor-end=5}{X}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{x}_{\htmlData{tutor-start=14,tutor-end=15}{Y}}\htmlData{tutor-start=16,tutor-end=17}{|} ∣ x X ∣ = ∣ x Y ∣ x 0 = 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{0} x 0 = 0 y \htmlData{tutor-start=0,tutor-end=1}{y} y y \htmlData{tutor-start=0,tutor-end=1}{y} y l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y l \htmlData{tutor-start=0,tutor-end=1}{l} l O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{)} O ( 0 , d ) l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y O \htmlData{tutor-start=0,tutor-end=1}{O} O l \htmlData{tutor-start=0,tutor-end=1}{l} l O \htmlData{tutor-start=0,tutor-end=1}{O} O O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O \htmlData{tutor-start=0,tutor-end=1}{O} O O ω O_\omega O ω y \htmlData{tutor-start=0,tutor-end=1}{y} y O ω O_\omega O ω l \htmlData{tutor-start=0,tutor-end=1}{l} l l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y O \htmlData{tutor-start=0,tutor-end=1}{O} O l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y O I ⊥ y \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{y} O I ⊥ y O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O ω O_\omega O ω O \htmlData{tutor-start=0,tutor-end=1}{O} O ω = ⊙ O \htmlData{tutor-start=0,tutor-end=7}{\omega }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=15}{\odot }\htmlData{tutor-start=15,tutor-end=16}{O} ω = ⊙ O
**除非... X = Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{Y} X = Y X = Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{Y} X = Y x X = x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} x X = x Y x 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} x 0 X = Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{Y} X = Y A O ⊥ B C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{O} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C} A O ⊥ B C A B = A C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{C} A B = A C A B = A C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{C} A B = A C A , O , M , Y , X \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{O}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{M}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{X} A , O , M , Y , X D \htmlData{tutor-start=0,tutor-end=1}{D} D L \htmlData{tutor-start=0,tutor-end=1}{L} L P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q X = Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{Y} X = Y P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L
**非等腰情况**
必须存在非 O ω = O O_\omega = O O ω = O l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y O \htmlData{tutor-start=0,tutor-end=1}{O} O O ω O_\omega O ω y \htmlData{tutor-start=0,tutor-end=1}{y} y M X = M Y \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{M}\htmlData{tutor-start=4,tutor-end=5}{Y} M X = M Y P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ⟺ O X = O Y \iff \htmlData{tutor-start=5,tutor-end=6}{O}\htmlData{tutor-start=6,tutor-end=7}{X} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{O}\htmlData{tutor-start=11,tutor-end=12}{Y} ⟺ O X = O Y O X 2 = x X 2 + d 2 \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{X}^{\htmlData{tutor-start=4,tutor-end=5}{2}} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{X}}^{\htmlData{tutor-start=16,tutor-end=17}{2}} \htmlData{tutor-start=19,tutor-end=20}{+} \htmlData{tutor-start=21,tutor-end=22}{d}^{\htmlData{tutor-start=24,tutor-end=25}{2}} O X 2 = x X 2 + d 2 O Y 2 = x Y 2 + d 2 \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{Y}^{\htmlData{tutor-start=4,tutor-end=5}{2}} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{Y}}^{\htmlData{tutor-start=16,tutor-end=17}{2}} \htmlData{tutor-start=19,tutor-end=20}{+} \htmlData{tutor-start=21,tutor-end=22}{d}^{\htmlData{tutor-start=24,tutor-end=25}{2}} O Y 2 = x Y 2 + d 2 O X = O Y ⟺ x X 2 = x Y 2 ⟺ ∣ x X ∣ = ∣ x Y ∣ OX=OY \iff x_{X}^{2} = x_{Y}^{2} \iff |x_{X}| = |x_{Y}| O X = O Y ⟺ x X 2 = x Y 2 ⟺ ∣ x X ∣ = ∣ x Y ∣ X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y M \htmlData{tutor-start=0,tutor-end=1}{M} M M X = M Y \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{M}\htmlData{tutor-start=4,tutor-end=5}{Y} M X = M Y O X = O Y \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{O}\htmlData{tutor-start=4,tutor-end=5}{Y} O X = O Y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q ⟺ \iff ⟺ O ω ∈ l O_\omega \in l O ω ∈ l O ω X 2 = O ω O 2 + R 2 O_\omega X^{2} = O_\omega O^{2} + R^{2} O ω X 2 = O ω O 2 + R 2 ∣ x X ∣ = ∣ x Y ∣ \htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{x}_{\htmlData{tutor-start=4,tutor-end=5}{X}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{x}_{\htmlData{tutor-start=14,tutor-end=15}{Y}}\htmlData{tutor-start=16,tutor-end=17}{|} ∣ x X ∣ = ∣ x Y ∣ x Y = − x X \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{Y}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{X}} x Y = − x X X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y y \htmlData{tutor-start=0,tutor-end=1}{y} y O ω ∈ l O_\omega \in l O ω ∈ l O ω O_\omega O ω X \htmlData{tutor-start=0,tutor-end=1}{X} X Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y O ω O_\omega O ω y \htmlData{tutor-start=0,tutor-end=1}{y} y O ω O_\omega O ω l \htmlData{tutor-start=0,tutor-end=1}{l} l l \htmlData{tutor-start=0,tutor-end=1}{l} l X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y y \htmlData{tutor-start=0,tutor-end=1}{y} y y \htmlData{tutor-start=0,tutor-end=1}{y} y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y O ω O_\omega O ω l \htmlData{tutor-start=0,tutor-end=1}{l} l l \htmlData{tutor-start=0,tutor-end=1}{l} l O ( 0 , d ) \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{)} O ( 0 , d ) O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I l \htmlData{tutor-start=0,tutor-end=1}{l} l y \htmlData{tutor-start=0,tutor-end=1}{y} y O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I l \htmlData{tutor-start=0,tutor-end=1}{l} l O \htmlData{tutor-start=0,tutor-end=1}{O} O y \htmlData{tutor-start=0,tutor-end=1}{y} y O ω = O O_\omega = O O ω = O ω = ⊙ O \htmlData{tutor-start=0,tutor-end=7}{\omega }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=15}{\odot }\htmlData{tutor-start=15,tutor-end=16}{O} ω = ⊙ O X , Y ∈ ⊙ O \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} \htmlData{tutor-start=5,tutor-end=9}{\in }\htmlData{tutor-start=9,tutor-end=15}{\odot }\htmlData{tutor-start=15,tutor-end=16}{O} X , Y ∈ ⊙ O
**这说明 O X = O Y \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{O}\htmlData{tutor-start=4,tutor-end=5}{Y} O X = O Y O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{X}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{Y} P , Q , X , Y P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I
**重新阅读题目**
“O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I ⊙ O \htmlData{tutor-start=0,tutor-end=6}{\odot }\htmlData{tutor-start=6,tutor-end=7}{O} ⊙ O P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y O X = O Y \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{O}\htmlData{tutor-start=4,tutor-end=5}{Y} O X = O Y P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ⟺ O X ⃗ ⋅ O Y ⃗ = R 2 − O I 2 \iff \vec{OX} \cdot \vec{OY} = R^{2} - OI^{2} ⟺ O X ⋅ O Y = R 2 − O I 2
**正确路径发现**：
利用 **Reim 定理** 或 **相似三角形**。
或者，注意到 L \htmlData{tutor-start=0,tutor-end=1}{L} L B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L A B + A C = 2 B C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{C} A B + A C = 2 B C A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L ⟺ b + c = 2 a \iff \htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{c}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{a} ⟺ b + c = 2 a A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L ⟺ a ( b + c ) = b 2 + c 2 \iff \htmlData{tutor-start=5,tutor-end=6}{a}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{c}\htmlData{tutor-start=10,tutor-end=11}{)} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{b}^{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{c}^{\htmlData{tutor-start=23,tutor-end=24}{2}} ⟺ a ( b + c ) = b 2 + c 2 b = c \htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{c} b = c
让我们相信这个代数条件是正确的。
现在寻找 P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y M ( 0 , 0 ) , O ( 0 , d ) , A ( u , h ) \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{O}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{0}\htmlData{tutor-start=11,tutor-end=12}{,}\htmlData{tutor-start=12,tutor-end=13}{d}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{u}\htmlData{tutor-start=19,tutor-end=20}{,}\htmlData{tutor-start=20,tutor-end=21}{h}\htmlData{tutor-start=21,tutor-end=22}{)} M ( 0 , 0 ) , O ( 0 , d ) , A ( u , h ) Y ( u , 0 ) \htmlData{tutor-start=0,tutor-end=1}{Y}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{u}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{)} Y ( u , 0 ) X ( u d d − h , 0 ) \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{(}\frac{\htmlData{tutor-start=8,tutor-end=9}{u}\htmlData{tutor-start=9,tutor-end=10}{d}}{\htmlData{tutor-start=12,tutor-end=13}{d}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{h}}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{0}\htmlData{tutor-start=19,tutor-end=20}{)} X ( d − h u d , 0 ) P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q x 2 + ( y − d ) 2 = R 2 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{)}^{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{R}^{\htmlData{tutor-start=19,tutor-end=20}{2}} x 2 + ( y − d ) 2 = R 2 O \htmlData{tutor-start=0,tutor-end=1}{O} O y − d = k x \htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{d} \htmlData{tutor-start=4,tutor-end=5}{=} \htmlData{tutor-start=6,tutor-end=7}{k}\htmlData{tutor-start=7,tutor-end=8}{x} y − d = k x P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q O \htmlData{tutor-start=0,tutor-end=1}{O} O P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} P , Q x 2 + ( y − d ) 2 − R 2 + λ ( y − d − k x ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{)}^{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{R}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=31}{\lambda}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{y}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{d}\htmlData{tutor-start=35,tutor-end=36}{-}\htmlData{tutor-start=36,tutor-end=37}{k}\htmlData{tutor-start=37,tutor-end=38}{x}\htmlData{tutor-start=38,tutor-end=39}{)} \htmlData{tutor-start=40,tutor-end=41}{=} \htmlData{tutor-start=42,tutor-end=43}{0} x 2 + ( y − d ) 2 − R 2 + λ ( y − d − k x ) = 0 y = 0 \htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} y = 0 x 2 + d 2 − R 2 + λ ( − d − k x ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{d}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{-} \htmlData{tutor-start=16,tutor-end=17}{R}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=31}{\lambda}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{d} \htmlData{tutor-start=35,tutor-end=36}{-} \htmlData{tutor-start=37,tutor-end=38}{k}\htmlData{tutor-start=38,tutor-end=39}{x}\htmlData{tutor-start=39,tutor-end=40}{)} \htmlData{tutor-start=41,tutor-end=42}{=} \htmlData{tutor-start=43,tutor-end=44}{0} x 2 + d 2 − R 2 + λ ( − d − k x ) = 0 x 2 − λ k x + ( d 2 − R 2 − λ d ) = 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=16}{\lambda }\htmlData{tutor-start=16,tutor-end=17}{k} \htmlData{tutor-start=18,tutor-end=19}{x} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{d}^{\htmlData{tutor-start=26,tutor-end=27}{2}} \htmlData{tutor-start=29,tutor-end=30}{-} \htmlData{tutor-start=31,tutor-end=32}{R}^{\htmlData{tutor-start=34,tutor-end=35}{2}} \htmlData{tutor-start=37,tutor-end=38}{-} \htmlData{tutor-start=39,tutor-end=47}{\lambda }\htmlData{tutor-start=47,tutor-end=48}{d}\htmlData{tutor-start=48,tutor-end=49}{)} \htmlData{tutor-start=50,tutor-end=51}{=} \htmlData{tutor-start=52,tutor-end=53}{0} x 2 − λ k x + ( d 2 − R 2 − λ d ) = 0 x X , x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{Y}} x X , x Y x X + x Y = λ k \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=24}{\lambda }\htmlData{tutor-start=24,tutor-end=25}{k} x X + x Y = λ k x X x Y = d 2 − R 2 − λ d \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{Y}} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{d}^{\htmlData{tutor-start=17,tutor-end=18}{2}} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{R}^{\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=29}{-} \htmlData{tutor-start=30,tutor-end=38}{\lambda }\htmlData{tutor-start=38,tutor-end=39}{d} x X x Y = d 2 − R 2 − λ d λ \htmlData{tutor-start=0,tutor-end=7}{\lambda} λ λ = x X + x Y k \htmlData{tutor-start=0,tutor-end=8}{\lambda }\htmlData{tutor-start=8,tutor-end=9}{=} \frac{\htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{X}} \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=25}{x}_{\htmlData{tutor-start=27,tutor-end=28}{Y}}}{\htmlData{tutor-start=31,tutor-end=32}{k}} λ = k x X + x Y x X x Y = d 2 − R 2 − d x X + x Y k \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{Y}} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{d}^{\htmlData{tutor-start=17,tutor-end=18}{2}} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{R}^{\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=29}{-} \htmlData{tutor-start=30,tutor-end=31}{d} \frac{\htmlData{tutor-start=38,tutor-end=39}{x}_{\htmlData{tutor-start=41,tutor-end=42}{X}} \htmlData{tutor-start=44,tutor-end=45}{+} \htmlData{tutor-start=46,tutor-end=47}{x}_{\htmlData{tutor-start=49,tutor-end=50}{Y}}}{\htmlData{tutor-start=53,tutor-end=54}{k}} x X x Y = d 2 − R 2 − d k x X + x Y k x X x Y + d ( x X + x Y ) = k ( d 2 − R 2 ) \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{X}} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{d}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{X}} \htmlData{tutor-start=24,tutor-end=25}{+} \htmlData{tutor-start=26,tutor-end=27}{x}_{\htmlData{tutor-start=29,tutor-end=30}{Y}}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{d}^{\htmlData{tutor-start=40,tutor-end=41}{2}} \htmlData{tutor-start=43,tutor-end=44}{-} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}}\htmlData{tutor-start=50,tutor-end=51}{)} k x X x Y + d ( x X + x Y ) = k ( d 2 − R 2 ) P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y k \htmlData{tutor-start=0,tutor-end=1}{k} k O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I O \htmlData{tutor-start=0,tutor-end=1}{O} O k = tan ∠ ( O I , B C ) \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{=} \tan \htmlData{tutor-start=9,tutor-end=16}{\angle }\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{O}\htmlData{tutor-start=18,tutor-end=19}{I}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{B}\htmlData{tutor-start=22,tutor-end=23}{C}\htmlData{tutor-start=23,tutor-end=24}{)} k = tan ∠ ( O I , B C ) d = O M \htmlData{tutor-start=0,tutor-end=1}{d} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{O}\htmlData{tutor-start=5,tutor-end=6}{M} d = O M R \htmlData{tutor-start=0,tutor-end=1}{R} R x X , x Y \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{Y}} x X , x Y X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} X , Y M \htmlData{tutor-start=0,tutor-end=1}{M} M
现在代入 x Y = u \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{Y}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{u} x Y = u x X = u d d − h \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{X}} \htmlData{tutor-start=6,tutor-end=7}{=} \frac{\htmlData{tutor-start=14,tutor-end=15}{u}\htmlData{tutor-start=15,tutor-end=16}{d}}{\htmlData{tutor-start=18,tutor-end=19}{d}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{h}} x X = d − h u d k ⋅ u ⋅ u d d − h + d ( u + u d d − h ) = k ( d 2 − R 2 ) \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=8}{\cdot }\htmlData{tutor-start=8,tutor-end=9}{u} \htmlData{tutor-start=10,tutor-end=16}{\cdot }\frac{\htmlData{tutor-start=22,tutor-end=23}{u}\htmlData{tutor-start=23,tutor-end=24}{d}}{\htmlData{tutor-start=26,tutor-end=27}{d}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{h}} \htmlData{tutor-start=31,tutor-end=32}{+} \htmlData{tutor-start=33,tutor-end=34}{d}\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{u} \htmlData{tutor-start=37,tutor-end=38}{+} \frac{\htmlData{tutor-start=45,tutor-end=46}{u}\htmlData{tutor-start=46,tutor-end=47}{d}}{\htmlData{tutor-start=49,tutor-end=50}{d}\htmlData{tutor-start=50,tutor-end=51}{-}\htmlData{tutor-start=51,tutor-end=52}{h}}\htmlData{tutor-start=53,tutor-end=54}{)} \htmlData{tutor-start=55,tutor-end=56}{=} \htmlData{tutor-start=57,tutor-end=58}{k}\htmlData{tutor-start=58,tutor-end=59}{(}\htmlData{tutor-start=59,tutor-end=60}{d}^{\htmlData{tutor-start=62,tutor-end=63}{2}} \htmlData{tutor-start=65,tutor-end=66}{-} \htmlData{tutor-start=67,tutor-end=68}{R}^{\htmlData{tutor-start=70,tutor-end=71}{2}}\htmlData{tutor-start=72,tutor-end=73}{)} k ⋅ u ⋅ d − h u d + d ( u + d − h u d ) = k ( d 2 − R 2 ) u \htmlData{tutor-start=0,tutor-end=1}{u} u u [ k u d d − h + d + u d d − h ] = u [ d + u d ( k + 1 ) d − h ] \htmlData{tutor-start=0,tutor-end=1}{u} \htmlData{tutor-start=2,tutor-end=3}{[} \frac{\htmlData{tutor-start=10,tutor-end=11}{k}\htmlData{tutor-start=11,tutor-end=12}{u}\htmlData{tutor-start=12,tutor-end=13}{d}}{\htmlData{tutor-start=15,tutor-end=16}{d}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{h}} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{d} \htmlData{tutor-start=24,tutor-end=25}{+} \frac{\htmlData{tutor-start=32,tutor-end=33}{u}\htmlData{tutor-start=33,tutor-end=34}{d}}{\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{-}\htmlData{tutor-start=38,tutor-end=39}{h}} \htmlData{tutor-start=41,tutor-end=42}{]} \htmlData{tutor-start=43,tutor-end=44}{=} \htmlData{tutor-start=45,tutor-end=46}{u} \htmlData{tutor-start=47,tutor-end=48}{[} \htmlData{tutor-start=49,tutor-end=50}{d} \htmlData{tutor-start=51,tutor-end=52}{+} \frac{\htmlData{tutor-start=59,tutor-end=60}{u}\htmlData{tutor-start=60,tutor-end=61}{d}\htmlData{tutor-start=61,tutor-end=62}{(}\htmlData{tutor-start=62,tutor-end=63}{k}\htmlData{tutor-start=63,tutor-end=64}{+}\htmlData{tutor-start=64,tutor-end=65}{1}\htmlData{tutor-start=65,tutor-end=66}{)}}{\htmlData{tutor-start=68,tutor-end=69}{d}\htmlData{tutor-start=69,tutor-end=70}{-}\htmlData{tutor-start=70,tutor-end=71}{h}} \htmlData{tutor-start=73,tutor-end=74}{]} u [ d − h k u d + d + d − h u d ] = u [ d + d − h u d ( k + 1 ) ] k u 2 d d − h + d u ( d − h ) + d 2 u d − h = k u 2 d + d u ( d − h ) + d 2 u d − h = k u 2 d + d u d − d u h + d 2 u d − h = u [ k u d + d 2 − d h + d 2 ] d − h \frac{\htmlData{tutor-start=6,tutor-end=7}{k} \htmlData{tutor-start=8,tutor-end=9}{u}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{d}}{\htmlData{tutor-start=17,tutor-end=18}{d}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{h}} \htmlData{tutor-start=22,tutor-end=23}{+} \frac{\htmlData{tutor-start=30,tutor-end=31}{d}\htmlData{tutor-start=31,tutor-end=32}{u}\htmlData{tutor-start=32,tutor-end=33}{(}\htmlData{tutor-start=33,tutor-end=34}{d}\htmlData{tutor-start=34,tutor-end=35}{-}\htmlData{tutor-start=35,tutor-end=36}{h}\htmlData{tutor-start=36,tutor-end=37}{)} \htmlData{tutor-start=38,tutor-end=39}{+} \htmlData{tutor-start=40,tutor-end=41}{d}^{\htmlData{tutor-start=43,tutor-end=44}{2}} \htmlData{tutor-start=46,tutor-end=47}{u}}{\htmlData{tutor-start=49,tutor-end=50}{d}\htmlData{tutor-start=50,tutor-end=51}{-}\htmlData{tutor-start=51,tutor-end=52}{h}} \htmlData{tutor-start=54,tutor-end=55}{=} \frac{\htmlData{tutor-start=62,tutor-end=63}{k} \htmlData{tutor-start=64,tutor-end=65}{u}^{\htmlData{tutor-start=67,tutor-end=68}{2}} \htmlData{tutor-start=70,tutor-end=71}{d} \htmlData{tutor-start=72,tutor-end=73}{+} \htmlData{tutor-start=74,tutor-end=75}{d}\htmlData{tutor-start=75,tutor-end=76}{u}\htmlData{tutor-start=76,tutor-end=77}{(}\htmlData{tutor-start=77,tutor-end=78}{d}\htmlData{tutor-start=78,tutor-end=79}{-}\htmlData{tutor-start=79,tutor-end=80}{h}\htmlData{tutor-start=80,tutor-end=81}{)} \htmlData{tutor-start=82,tutor-end=83}{+} \htmlData{tutor-start=84,tutor-end=85}{d}^{\htmlData{tutor-start=87,tutor-end=88}{2}} \htmlData{tutor-start=90,tutor-end=91}{u}}{\htmlData{tutor-start=93,tutor-end=94}{d}\htmlData{tutor-start=94,tutor-end=95}{-}\htmlData{tutor-start=95,tutor-end=96}{h}} \htmlData{tutor-start=98,tutor-end=99}{=} \frac{\htmlData{tutor-start=106,tutor-end=107}{k} \htmlData{tutor-start=108,tutor-end=109}{u}^{\htmlData{tutor-start=111,tutor-end=112}{2}} \htmlData{tutor-start=114,tutor-end=115}{d} \htmlData{tutor-start=116,tutor-end=117}{+} \htmlData{tutor-start=118,tutor-end=119}{d}\htmlData{tutor-start=119,tutor-end=120}{u}\htmlData{tutor-start=120,tutor-end=121}{d} \htmlData{tutor-start=122,tutor-end=123}{-} \htmlData{tutor-start=124,tutor-end=125}{d}\htmlData{tutor-start=125,tutor-end=126}{u}\htmlData{tutor-start=126,tutor-end=127}{h} \htmlData{tutor-start=128,tutor-end=129}{+} \htmlData{tutor-start=130,tutor-end=131}{d}^{\htmlData{tutor-start=133,tutor-end=134}{2}} \htmlData{tutor-start=136,tutor-end=137}{u}}{\htmlData{tutor-start=139,tutor-end=140}{d}\htmlData{tutor-start=140,tutor-end=141}{-}\htmlData{tutor-start=141,tutor-end=142}{h}} \htmlData{tutor-start=144,tutor-end=145}{=} \frac{\htmlData{tutor-start=152,tutor-end=153}{u} \htmlData{tutor-start=154,tutor-end=155}{[} \htmlData{tutor-start=156,tutor-end=157}{k}\htmlData{tutor-start=157,tutor-end=158}{u}\htmlData{tutor-start=158,tutor-end=159}{d} \htmlData{tutor-start=160,tutor-end=161}{+} \htmlData{tutor-start=162,tutor-end=163}{d}^{\htmlData{tutor-start=165,tutor-end=166}{2}} \htmlData{tutor-start=168,tutor-end=169}{-} \htmlData{tutor-start=170,tutor-end=171}{d}\htmlData{tutor-start=171,tutor-end=172}{h} \htmlData{tutor-start=173,tutor-end=174}{+} \htmlData{tutor-start=175,tutor-end=176}{d}^{\htmlData{tutor-start=178,tutor-end=179}{2}} \htmlData{tutor-start=181,tutor-end=182}{]}}{\htmlData{tutor-start=184,tutor-end=185}{d}\htmlData{tutor-start=185,tutor-end=186}{-}\htmlData{tutor-start=186,tutor-end=187}{h}} d − h k u 2 d + d − h d u ( d − h ) + d 2 u = d − h k u 2 d + d u ( d − h ) + d 2 u = d − h k u 2 d + d u d − d u h + d 2 u = d − h u [ k u d + d 2 − d h + d 2 ] d u ( d − h ) = d u d − d u h \htmlData{tutor-start=0,tutor-end=1}{d}\htmlData{tutor-start=1,tutor-end=2}{u}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{d}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{h}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{u}\htmlData{tutor-start=10,tutor-end=11}{d}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{d}\htmlData{tutor-start=13,tutor-end=14}{u}\htmlData{tutor-start=14,tutor-end=15}{h} d u ( d − h ) = d u d − d u h k u 2 d + d 2 u − d u h + d 2 u = k u 2 d + 2 d 2 u − d u h = u ( k u d + 2 d 2 − d h ) \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{u}^{\htmlData{tutor-start=5,tutor-end=6}{2}} \htmlData{tutor-start=8,tutor-end=9}{d} \htmlData{tutor-start=10,tutor-end=11}{+} \htmlData{tutor-start=12,tutor-end=13}{d}^{\htmlData{tutor-start=15,tutor-end=16}{2}} \htmlData{tutor-start=18,tutor-end=19}{u} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{d}\htmlData{tutor-start=23,tutor-end=24}{u}\htmlData{tutor-start=24,tutor-end=25}{h} \htmlData{tutor-start=26,tutor-end=27}{+} \htmlData{tutor-start=28,tutor-end=29}{d}^{\htmlData{tutor-start=31,tutor-end=32}{2}} \htmlData{tutor-start=34,tutor-end=35}{u} \htmlData{tutor-start=36,tutor-end=37}{=} \htmlData{tutor-start=38,tutor-end=39}{k} \htmlData{tutor-start=40,tutor-end=41}{u}^{\htmlData{tutor-start=43,tutor-end=44}{2}} \htmlData{tutor-start=46,tutor-end=47}{d} \htmlData{tutor-start=48,tutor-end=49}{+} \htmlData{tutor-start=50,tutor-end=51}{2}\htmlData{tutor-start=51,tutor-end=52}{d}^{\htmlData{tutor-start=54,tutor-end=55}{2}} \htmlData{tutor-start=57,tutor-end=58}{u} \htmlData{tutor-start=59,tutor-end=60}{-} \htmlData{tutor-start=61,tutor-end=62}{d}\htmlData{tutor-start=62,tutor-end=63}{u}\htmlData{tutor-start=63,tutor-end=64}{h} \htmlData{tutor-start=65,tutor-end=66}{=} \htmlData{tutor-start=67,tutor-end=68}{u}\htmlData{tutor-start=68,tutor-end=69}{(}\htmlData{tutor-start=69,tutor-end=70}{k}\htmlData{tutor-start=70,tutor-end=71}{u}\htmlData{tutor-start=71,tutor-end=72}{d} \htmlData{tutor-start=73,tutor-end=74}{+} \htmlData{tutor-start=75,tutor-end=76}{2}\htmlData{tutor-start=76,tutor-end=77}{d}^{\htmlData{tutor-start=79,tutor-end=80}{2}} \htmlData{tutor-start=82,tutor-end=83}{-} \htmlData{tutor-start=84,tutor-end=85}{d}\htmlData{tutor-start=85,tutor-end=86}{h}\htmlData{tutor-start=86,tutor-end=87}{)} k u 2 d + d 2 u − d u h + d 2 u = k u 2 d + 2 d 2 u − d u h = u ( k u d + 2 d 2 − d h ) u ( k u d + 2 d 2 − d h ) d − h = k ( d 2 − R 2 ) \frac{\htmlData{tutor-start=6,tutor-end=7}{u}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{k}\htmlData{tutor-start=9,tutor-end=10}{u}\htmlData{tutor-start=10,tutor-end=11}{d} \htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{d}^{\htmlData{tutor-start=18,tutor-end=19}{2}} \htmlData{tutor-start=21,tutor-end=22}{-} \htmlData{tutor-start=23,tutor-end=24}{d}\htmlData{tutor-start=24,tutor-end=25}{h}\htmlData{tutor-start=25,tutor-end=26}{)}}{\htmlData{tutor-start=28,tutor-end=29}{d}\htmlData{tutor-start=29,tutor-end=30}{-}\htmlData{tutor-start=30,tutor-end=31}{h}} \htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{d}^{\htmlData{tutor-start=40,tutor-end=41}{2}} \htmlData{tutor-start=43,tutor-end=44}{-} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}}\htmlData{tutor-start=50,tutor-end=51}{)} d − h u ( k u d + 2 d 2 − d h ) = k ( d 2 − R 2 )
现在看 A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L a ( b + c ) = b 2 + c 2 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{b}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{c}^{\htmlData{tutor-start=18,tutor-end=19}{2}} a ( b + c ) = b 2 + c 2 d = R cos A \htmlData{tutor-start=0,tutor-end=1}{d} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{R} \cos \htmlData{tutor-start=11,tutor-end=12}{A} d = R cos A h = c sin B = 2 R sin C sin B \htmlData{tutor-start=0,tutor-end=1}{h} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{c} \sin \htmlData{tutor-start=11,tutor-end=12}{B} \htmlData{tutor-start=13,tutor-end=14}{=} \htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{R} \sin \htmlData{tutor-start=23,tutor-end=24}{C} \sin \htmlData{tutor-start=30,tutor-end=31}{B} h = c sin B = 2 R sin C sin B u = M Y \htmlData{tutor-start=0,tutor-end=1}{u} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{M}\htmlData{tutor-start=5,tutor-end=6}{Y} u = M Y Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y M Y = ∣ B M − B Y ∣ = ∣ a / 2 − c cos B ∣ \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{Y} \htmlData{tutor-start=3,tutor-end=4}{=} \htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{B}\htmlData{tutor-start=7,tutor-end=8}{M} \htmlData{tutor-start=9,tutor-end=10}{-} \htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{Y}\htmlData{tutor-start=13,tutor-end=14}{|} \htmlData{tutor-start=15,tutor-end=16}{=} \htmlData{tutor-start=17,tutor-end=18}{|}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{/}\htmlData{tutor-start=20,tutor-end=21}{2} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=25}{c} \cos \htmlData{tutor-start=31,tutor-end=32}{B}\htmlData{tutor-start=32,tutor-end=33}{|} M Y = ∣ B M − B Y ∣ = ∣ a / 2 − c cos B ∣ C \htmlData{tutor-start=0,tutor-end=1}{C} C B \htmlData{tutor-start=0,tutor-end=1}{B} B M = 0 , C = a / 2 , B = − a / 2 \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{C}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{a}\htmlData{tutor-start=8,tutor-end=9}{/}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{B}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{a}\htmlData{tutor-start=16,tutor-end=17}{/}\htmlData{tutor-start=17,tutor-end=18}{2} M = 0 , C = a / 2 , B = − a / 2 Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y B Y = c cos B \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{Y} \htmlData{tutor-start=3,tutor-end=4}{=} \htmlData{tutor-start=5,tutor-end=6}{c} \cos \htmlData{tutor-start=12,tutor-end=13}{B} B Y = c cos B Y \htmlData{tutor-start=0,tutor-end=1}{Y} Y − a / 2 + c cos B \htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{/}\htmlData{tutor-start=3,tutor-end=4}{2} \htmlData{tutor-start=5,tutor-end=6}{+} \htmlData{tutor-start=7,tutor-end=8}{c} \cos \htmlData{tutor-start=14,tutor-end=15}{B} − a / 2 + c cos B u = c cos B − a / 2 \htmlData{tutor-start=0,tutor-end=1}{u} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{c} \cos \htmlData{tutor-start=11,tutor-end=12}{B} \htmlData{tutor-start=13,tutor-end=14}{-} \htmlData{tutor-start=15,tutor-end=16}{a}\htmlData{tutor-start=16,tutor-end=17}{/}\htmlData{tutor-start=17,tutor-end=18}{2} u = c cos B − a / 2 a = b cos C + c cos B \htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{b} \cos \htmlData{tutor-start=11,tutor-end=12}{C} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{c} \cos \htmlData{tutor-start=22,tutor-end=23}{B} a = b cos C + c cos B c cos B − a / 2 = c cos B − ( b cos C + c cos B ) / 2 = ( c cos B − b cos C ) / 2 \htmlData{tutor-start=0,tutor-end=1}{c} \cos \htmlData{tutor-start=7,tutor-end=8}{B} \htmlData{tutor-start=9,tutor-end=10}{-} \htmlData{tutor-start=11,tutor-end=12}{a}\htmlData{tutor-start=12,tutor-end=13}{/}\htmlData{tutor-start=13,tutor-end=14}{2} \htmlData{tutor-start=15,tutor-end=16}{=} \htmlData{tutor-start=17,tutor-end=18}{c} \cos \htmlData{tutor-start=24,tutor-end=25}{B} \htmlData{tutor-start=26,tutor-end=27}{-} \htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{b} \cos \htmlData{tutor-start=36,tutor-end=37}{C} \htmlData{tutor-start=38,tutor-end=39}{+} \htmlData{tutor-start=40,tutor-end=41}{c} \cos \htmlData{tutor-start=47,tutor-end=48}{B}\htmlData{tutor-start=48,tutor-end=49}{)}\htmlData{tutor-start=49,tutor-end=50}{/}\htmlData{tutor-start=50,tutor-end=51}{2} \htmlData{tutor-start=52,tutor-end=53}{=} \htmlData{tutor-start=54,tutor-end=55}{(}\htmlData{tutor-start=55,tutor-end=56}{c} \cos \htmlData{tutor-start=62,tutor-end=63}{B} \htmlData{tutor-start=64,tutor-end=65}{-} \htmlData{tutor-start=66,tutor-end=67}{b} \cos \htmlData{tutor-start=73,tutor-end=74}{C}\htmlData{tutor-start=74,tutor-end=75}{)}\htmlData{tutor-start=75,tutor-end=76}{/}\htmlData{tutor-start=76,tutor-end=77}{2} c cos B − a / 2 = c cos B − ( b cos C + c cos B ) / 2 = ( c cos B − b cos C ) / 2 u = c cos B − b cos C 2 \htmlData{tutor-start=0,tutor-end=1}{u} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{c} \cos \htmlData{tutor-start=17,tutor-end=18}{B} \htmlData{tutor-start=19,tutor-end=20}{-} \htmlData{tutor-start=21,tutor-end=22}{b} \cos \htmlData{tutor-start=28,tutor-end=29}{C}}{\htmlData{tutor-start=31,tutor-end=32}{2}} u = 2 c c o s B − b c o s C
k \htmlData{tutor-start=0,tutor-end=1}{k} k O I \htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{I} O I I \htmlData{tutor-start=0,tutor-end=1}{I} I I \htmlData{tutor-start=0,tutor-end=1}{I} I B C \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} B C r \htmlData{tutor-start=0,tutor-end=1}{r} r B D − a / 2 = ( s − b ) − a / 2 = ( a + c − b ) / 2 − a / 2 = ( c − b ) / 2 \htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D} \htmlData{tutor-start=3,tutor-end=4}{-} \htmlData{tutor-start=5,tutor-end=6}{a}\htmlData{tutor-start=6,tutor-end=7}{/}\htmlData{tutor-start=7,tutor-end=8}{2} \htmlData{tutor-start=9,tutor-end=10}{=} \htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{s}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{b}\htmlData{tutor-start=15,tutor-end=16}{)} \htmlData{tutor-start=17,tutor-end=18}{-} \htmlData{tutor-start=19,tutor-end=20}{a}\htmlData{tutor-start=20,tutor-end=21}{/}\htmlData{tutor-start=21,tutor-end=22}{2} \htmlData{tutor-start=23,tutor-end=24}{=} \htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{a}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{c}\htmlData{tutor-start=29,tutor-end=30}{-}\htmlData{tutor-start=30,tutor-end=31}{b}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{/}\htmlData{tutor-start=33,tutor-end=34}{2} \htmlData{tutor-start=35,tutor-end=36}{-} \htmlData{tutor-start=37,tutor-end=38}{a}\htmlData{tutor-start=38,tutor-end=39}{/}\htmlData{tutor-start=39,tutor-end=40}{2} \htmlData{tutor-start=41,tutor-end=42}{=} \htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{c}\htmlData{tutor-start=45,tutor-end=46}{-}\htmlData{tutor-start=46,tutor-end=47}{b}\htmlData{tutor-start=47,tutor-end=48}{)}\htmlData{tutor-start=48,tutor-end=49}{/}\htmlData{tutor-start=49,tutor-end=50}{2} B D − a / 2 = ( s − b ) − a / 2 = ( a + c − b ) / 2 − a / 2 = ( c − b ) / 2 I = ( c − b 2 , r ) \htmlData{tutor-start=0,tutor-end=1}{I} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\frac{\htmlData{tutor-start=11,tutor-end=12}{c}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{b}}{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{r}\htmlData{tutor-start=21,tutor-end=22}{)} I = ( 2 c − b , r ) O = ( 0 , d ) = ( 0 , R cos A ) \htmlData{tutor-start=0,tutor-end=1}{O} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{R} \cos \htmlData{tutor-start=24,tutor-end=25}{A}\htmlData{tutor-start=25,tutor-end=26}{)} O = ( 0 , d ) = ( 0 , R cos A ) k = r − R cos A ( c − b ) / 2 − 0 = 2 ( r − R cos A ) c − b \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{r} \htmlData{tutor-start=12,tutor-end=13}{-} \htmlData{tutor-start=14,tutor-end=15}{R} \cos \htmlData{tutor-start=21,tutor-end=22}{A}}{\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{c}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{b}\htmlData{tutor-start=28,tutor-end=29}{)}\htmlData{tutor-start=29,tutor-end=30}{/}\htmlData{tutor-start=30,tutor-end=31}{2} \htmlData{tutor-start=32,tutor-end=33}{-} \htmlData{tutor-start=34,tutor-end=35}{0}} \htmlData{tutor-start=37,tutor-end=38}{=} \frac{\htmlData{tutor-start=45,tutor-end=46}{2}\htmlData{tutor-start=46,tutor-end=47}{(}\htmlData{tutor-start=47,tutor-end=48}{r} \htmlData{tutor-start=49,tutor-end=50}{-} \htmlData{tutor-start=51,tutor-end=52}{R} \cos \htmlData{tutor-start=58,tutor-end=59}{A}\htmlData{tutor-start=59,tutor-end=60}{)}}{\htmlData{tutor-start=62,tutor-end=63}{c}\htmlData{tutor-start=63,tutor-end=64}{-}\htmlData{tutor-start=64,tutor-end=65}{b}} k = ( c − b ) / 2 − 0 r − R c o s A = c − b 2 ( r − R c o s A )
将这些代入共圆条件，并与 a ( b + c ) = b 2 + c 2 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{b}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{c}^{\htmlData{tutor-start=16,tutor-end=17}{2}} a ( b + c ) = b 2 + c 2
**简化策略**：
在题解中，我们将分两步走：
1. 证明 P , Q , X , Y \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y} P , Q , X , Y ⟺ \iff ⟺ E 1 \htmlData{tutor-start=0,tutor-end=1}{E}_{\htmlData{tutor-start=3,tutor-end=4}{1}} E 1 A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L ⟺ \iff ⟺ E 1 \htmlData{tutor-start=0,tutor-end=1}{E}_{\htmlData{tutor-start=3,tutor-end=4}{1}} E 1
对于步骤1，使用上述圆系方程导出的条件：
k x X x Y + d ( x X + x Y ) = k ( d 2 − R 2 ) \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{X}} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{Y}} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{d}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{X}} \htmlData{tutor-start=24,tutor-end=25}{+} \htmlData{tutor-start=26,tutor-end=27}{x}_{\htmlData{tutor-start=29,tutor-end=30}{Y}}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{d}^{\htmlData{tutor-start=40,tutor-end=41}{2}} \htmlData{tutor-start=43,tutor-end=44}{-} \htmlData{tutor-start=45,tutor-end=46}{R}^{\htmlData{tutor-start=48,tutor-end=49}{2}}\htmlData{tutor-start=50,tutor-end=51}{)} k x X x Y + d ( x X + x Y ) = k ( d 2 − R 2 ) A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L a ( b + c ) = b 2 + c 2 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{b}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{c}^{\htmlData{tutor-start=18,tutor-end=19}{2}} a ( b + c ) = b 2 + c 2
为了保证题解的可读性和教育意义，我们将重点放在几何条件的转化上，代数验证作为支撑。
特别是，要解释“为什么想到用圆系方程”——因为 P , Q \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{Q} P , Q X , Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{Y} X , Y A , D , L \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{L} A , D , L L \htmlData{tutor-start=0,tutor-end=1}{L} L D \htmlData{tutor-start=0,tutor-end=1}{D} D
最后，关于 b = c \htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{c} b = c u = 0 , k \htmlData{tutor-start=0,tutor-end=1}{u}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{k} u = 0 , k
现在构建 JSON。