[數學] 正三角形 :-o
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[數學] 正三角形 :-o
由 --- 於 星期二 五月 06, 2003 11:56 pm
試證: 單位圓上有三個相異點.如果此圓上任一點至此三點距離的乘積皆不大於 2,則證明此三點構成一個正三角形.
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--- - 訪客
由 Raceleader 於 星期三 五月 07, 2003 12:04 am
http://www.mathland.idv.tw/board/memo.asp?srcid=1682&bname=ASP
http://www.mathland.idv.tw/board/memo.asp?srcid=2602&bname=ASP
http://www.mathland.idv.tw/board/memo.asp?srcid=2602&bname=ASP
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Raceleader - 訪客
由 bugzpodder 於 星期三 十一月 05, 2003 10:40 am
Solution by Michael Lipnowski
Let ABC be such at triangle where MA*MB*MC<=2
Let A be the largest angle, and let the angle bisector of A intersect the circle at M. Suppose A >60, then we can find B' C' on the minor arc BC such that <MB'A=<MC'A=30, ie B'AC' is equilateral => MB'*MC'*MA=2
But MB'<MB, MC'<MC which implies MB*MC*MA >2, a contradiction
therefore the largest angle is at most 60 degrees => ABC is equilateral
Let ABC be such at triangle where MA*MB*MC<=2
Let A be the largest angle, and let the angle bisector of A intersect the circle at M. Suppose A >60, then we can find B' C' on the minor arc BC such that <MB'A=<MC'A=30, ie B'AC' is equilateral => MB'*MC'*MA=2
But MB'<MB, MC'<MC which implies MB*MC*MA >2, a contradiction
therefore the largest angle is at most 60 degrees => ABC is equilateral
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bugzpodder - 文章: 87
- 註冊時間: 2003-07-30
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