標題:
MQ31 --- Triangle
發問:
Difficulty: 40%In ?ABC,P, Q and R are points on BC, CA and AB such that AR/RB = BP/PC=CQ/QA = n. AP intersect BQ at X, BQ intersect CR at Yand CR intersectAP atZ. Express the ratio of area of ?ABC to that of ?XYZ interms of n. 更新: Can you further explain why △XBP = △YCQ = △ZAR ?
圖片參考：http://imgcld.yimg.com/8/n/HA04628698/o/701207120054913873412470.jpg △XBC / △XBA = CQ / QA [△XBP (n+1)/n] / △XBA = n △XBP / △XBA = n2 / (n+1)Let △XBP = n2 = △YCQ = △ZAR Then △XBA = n+1 = △YCB = △ZAC ∴ ?ABC = (?XYZ + △XBA + △YCB + △ZAC) = ?XYZ + 3(n+1) On the other hand , ?ABC = (△XBA + △XBP) (n+1)/n = (n2 + n+1) (n+1)/n ∴ ?ABC : ?XYZ = (n2+n+1)(n+1)/n : [(n2+n+1)(n+1)/n - 3(n+1)] = 1 : [1 - 3n/(n2+n+1)] = 1 : [(n2 -2n+1)/(n2+n+1)] = (n2 + n + 1) / (n - 1)2 2012-07-13 23:28:00 補充： Similarly , △YCQ / △YCB = n2 / (n+1) , therefore △YCQ / △BCQ = △XBP / △ABP , but △ABP = △BCQ = △ABC * n/(n+1) , thus △YCQ = △XBP. Similarly , △XBP = △YCQ = △ZAR.
其他解答:
MQ31 --- Triangle
發問:
Difficulty: 40%In ?ABC,P, Q and R are points on BC, CA and AB such that AR/RB = BP/PC=CQ/QA = n. AP intersect BQ at X, BQ intersect CR at Yand CR intersectAP atZ. Express the ratio of area of ?ABC to that of ?XYZ interms of n. 更新: Can you further explain why △XBP = △YCQ = △ZAR ?
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最佳解答:圖片參考：http://imgcld.yimg.com/8/n/HA04628698/o/701207120054913873412470.jpg △XBC / △XBA = CQ / QA [△XBP (n+1)/n] / △XBA = n △XBP / △XBA = n2 / (n+1)Let △XBP = n2 = △YCQ = △ZAR Then △XBA = n+1 = △YCB = △ZAC ∴ ?ABC = (?XYZ + △XBA + △YCB + △ZAC) = ?XYZ + 3(n+1) On the other hand , ?ABC = (△XBA + △XBP) (n+1)/n = (n2 + n+1) (n+1)/n ∴ ?ABC : ?XYZ = (n2+n+1)(n+1)/n : [(n2+n+1)(n+1)/n - 3(n+1)] = 1 : [1 - 3n/(n2+n+1)] = 1 : [(n2 -2n+1)/(n2+n+1)] = (n2 + n + 1) / (n - 1)2 2012-07-13 23:28:00 補充： Similarly , △YCQ / △YCB = n2 / (n+1) , therefore △YCQ / △BCQ = △XBP / △ABP , but △ABP = △BCQ = △ABC * n/(n+1) , thus △YCQ = △XBP. Similarly , △XBP = △YCQ = △ZAR.
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