2n-gon

yma16 · 发表于 2005-8-13 19:36:08

In a convex 2n-gon, form another 2n-gon by connecting the middle points of the sides. Could you prove or disprove that any two opposite sides of the new 2n-gon are parallel?

husonghu · 发表于 2005-8-14 05:05:13

Thank you for your trust. I'll try my best.

yma16 · 发表于 2005-8-14 03:21:17

Thank you for the post. Thank you very much to be the manager of the board.

For this one, I can prove when n=2, we will get a parallelogram (a well known fact). When n>2, the anwser is negative and it is not very interesting to form a counterexample.

For the n-gon question, you can try induction to prove the first part. I do not know the answer to the sencond part.

Can you get 野菜花、独木桥 and other experts back. We miss and need them. WXC has lost old people, too.

husonghu · 发表于 2005-8-13 21:51:23

可惜我不是数学行家。convex唯一性的证明有些思路，但暂时还写不出来。concave那
部份，我觉得如果是一个点凹进的话，答案应该是n-1，不知对不对？如果是多个点
凹进的话，则我就觉得很复杂的，不知答案是什么了。

我建议，有些现在暂时还没被答的题，你到时可重新post。

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2n-gon

Thank you for your trust. I'll try my best.

回复：yma16, 你的题都很棒。谢谢你。你的n-gon那题让我想了不少时间，.....

yma16, 你的题都很棒。谢谢你。你的n-gon那题让我想了不少时间，.....