1974年美国奥赛题的另一证明（图）

野菜花 · 发表于 2005-11-14 22:29:17

原题:www.ddhw.com

三角形ABC中一点D，使得它与三顶点的张角都是120度(

等边三角形RQP中有一点离三顶点的距离分别为a, b, c, 边长为 x试证 x=u+v+w

证明:www.ddhw.com

将三角形BCD绕B点沿逆时针方向旋转60度到三角形BEF的位置，这样BDE 和BCF 都是等边三角形。那么DE=v，CF=BF=a, EF=w

所以A，D，E，F 共线

AF=u+v+wwww.ddhw.com

以AF为一边作等边三角形AFG，只要证明BG=b

我们来证明三角形BFG 和CFA 全等：

CF=BF=a， AF=FG，

故BG=AC=b

yaluzangbu · 发表于 2005-11-15 01:24:03

How do you know that x is unique?

野菜花 · 发表于 2005-11-15 02:10:28

The original question only showed two pictures, and said to prove x=u+v+w.

I think I should have interpreted the problem like following:

There is a equilateral triangle with the length of u+v+w in whch there is point such that the distances from the point to the three vertices are a,b,c respectively. As in the book, there three proofs, no one proved the uniqueness.

I think QFT's proof showed the uniqueness.

乱弹 · 发表于 2005-11-15 03:08:25

not written a proof. It was hard for me to draw a nice picture, so I gave up.

This problem is very cute.

野菜花 · 发表于 2005-11-15 03:25:39

it for you! I am glad you like the problem.

husonghu · 发表于 2005-11-15 04:25:43

欢迎雅鲁藏布江流到这里！

constant · 发表于 2005-11-15 04:31:02

I'm lazy too, why didn't you draw for me?

寒潭清 · 发表于 2005-11-15 14:37:42

野菜花 · 发表于 2005-11-15 20:58:20

feel it is too easy to be bothered or you are lazy to write or to draw.

constant · 发表于 2005-11-15 21:21:00

I never knew how to prove geometry problems, so I don't even read them . Now I need to draw pictures about the cube kissing problem.

野菜花 · 发表于 2005-11-15 23:02:03

20多个立方体够你画的。

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1974年美国奥赛题的另一证明（图）

回复：1974年美国奥赛题的另一证明（图）

Good question![@};-]

Nice proof and pretty picture! Sorry for having

I know you are lazy to draw the picture, so I did

欢迎雅鲁藏布江流到这里！[@};-][@};-][>:D<][>:D<]

I'm lazy too, why didn't you draw for me? [:((]

[:)][@};-][@};-][:-][:-]

I know if you don't touch the question, either you

回复：I know if you don't touch the question, either

20多个立方体够你画的。[:D)][:))]

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1974年美国奥赛题的另一证明（图）

回复：1974年美国奥赛题的另一证明（图）

Good question![@};-]

Nice proof and pretty picture! Sorry for having

I know you are lazy to draw the picture, so I did

欢迎雅鲁藏布江流到这里！[@};-][@};-][>:D<][>:D<]

I'm lazy too, why didn't you draw for me? [:((]

[:)][@};-][@};-][:-*][:-*]

I know if you don't touch the question, either you

回复：I know if you don't touch the question, either

20多个立方体够你画的。[:D)][:))]

浏览过的版块

[:)][@};-][@};-][:-][:-]