关于种树问题答案的一个小注（图）

怀疑1 · 发表于 2005-4-8 21:05:58

这个题目实际上我是先任意取定了11， 13， 5， 7 这四个点，再顺次画出其他点的。对于 13树9排这个答案正确大概没什么疑问。有意思的是，后来sean9991 出了 14 树 10 排那个问题后，我留意到 8， 12，和 4 可能是共线的。现在做了一点计算，确信如此。

11， 13， 5， 7 这几个点不能太特殊，否则最后 4 这个点就在无穷远去了。我们取如下坐标：

11 （0，0）

13 (1, 0)

5(0, 1)

7(4/5, 3/5)

则可以依次计算出：

9（4/7，3/7）

3(2, 0)

1(0, 3)

12(2/3, 0)

8(0, 3/5)

4(4, -3)

可以验证 12， 8， 4 是共线的。

对一般的取点，也可以类似证明。只要让 7 的坐标为待定（a, b) 就可以了。

sean9991 头一个答案里也有共线的三个点在一条未画出的直线上。

怀疑1 · 发表于 2005-4-8 21:47:36

依据对称性，6， 10 和 4 没有理由不共线。不用算也可以确信。

sean9991 的两个答案实际上只有一条线不同。即把过 1， 3 的线去掉，添上过 8， 12 （或 10， 6）的线，并把树2 移动一下位置即可。

怀疑1 · 发表于 2005-4-8 22:04:39

8， 6， 2 和 12， 10 ， 2。

真是奇妙。人还怀疑我是那个贴题者吧。

sean9991 · 发表于 2005-4-8 22:50:42

there is a simple answer

怀疑1 · 发表于 2005-4-8 23:08:28

实际上我的证明比较复杂，但是不用头脑想，只死算。如果想一下，证明还可以更简单。我已经很少用平面几何做了，像这样的，我都算。

其实有些线去掉后，图像很简单。具体说，只保留1， 3， 4， 8， 12， 11 和相关的线，是一个很简单的题目，也许在什么仿射几何等等里是常识吧。不过我不知道它是否有名字。知道请告诉我。谢谢。

记得张景中院士是用面积法证明的。确实只有几行。

怀疑1 · 发表于 2005-4-8 23:15:03

还要保留9。

sean9991 · 发表于 2005-4-8 23:47:20

The coincedence of these points is the critical point of this problem, that is why I found alternative answers. When you did not get it, you did not solve it. I just wait for you to come to this yourself.

怀疑1 · 发表于 2005-4-9 01:17:30

I am sorry I do not know what you mean by "this problem". Your 14-10 problem is solved already. I also solved it. Yes, I used a big assumption yesterday, but I have showed it is correct. Do you think my proof is wrong or something? I did not try to do it yesterday because you did not think this was a problem until today.

Well, you might have some simple reason, and some others might have other good reasons (this is possible because people have different viewpoints and the math God makes them meet eventually). I want to hear all these reasons instead of finding them myself. Like luantan said, it is just a game.

Have fun.

乱弹 · 发表于 2005-4-9 01:29:50

他在WXC贴了那个 13 棵树的题目和共线题目。

fzy · 发表于 2005-4-9 01:36:42

They have not seen the problem before I posted it here.

geometryfan · 发表于 2005-4-10 01:13:07

I do not think it is so simple. This is closely related to two big nontrivial theorems in geometry. Maybe it is easy for you guys to prove it (yes, the posted proof is easy ), but did either of you know it before hand? I do not see that. One of you thought or assumed that it was possible, and the other thought it was not likely true. Then one of you found it would always hold, and the other then said there was a simple reason. Too funny! Seems that you two high hands are having a small fight here and many other guys, maybe only me, is too far away from you two.

Please. If you know something, please spell it out. This would better this lovely forum. Thank you.

sean9991 · 发表于 2005-4-11 20:38:59

people used this thousands years ago.

Here is the hint: it you learn a little bit of paiting before

geometryfan · 发表于 2005-4-11 20:49:25

Come on. If it was so simple, you should have come up with the answer in the very beginning. But instead, you doubted the other guy's correct answer.

sean9991 · 发表于 2005-4-11 21:06:37

this is the point used to solve the problem i proposed for 14 trees, and the reason why i postponed the answer of alternative answer for 13 trees because it used the same argument. That arrogant guy thought he got the answer before coming to the point, and strictly speaking his answer is not correct (answer with an extra assumption?). Since he said any questions for 14 trees or more are uninteresting (and now we know there are more questions to follow), and those crap about post doc, brah, brah, I surely won't give him any easy time. Eventually, he found out the point, and felt it amazing. Now I think it's the time to tell him it's a simple principle in painting that all parallel lines come to a point on the horizon.

geometryfan · 发表于 2005-4-11 21:30:48

I did not know that he offended you. I am sorry. But it seems that he also felt offended.

Anyway, peace is good.

怀疑1 · 发表于 2005-4-11 23:24:55

我是不该做那两个评论，我道歉。可是你可以直接说呀。何必这样做呢？一直说我的假定不成立，后来又好像早就知道那个假定是成立的，岂不是前言不搭后语？现在坦白了是要给我 hard time, 真是少见了。并没有什么 hard time, 我不会为做不出来一个题目伤心的。要是那样，fzy 那些题目让我都不敢来这儿玩了。

至于我有没有找到那个所谓的简单理由，我想说到无穷远点和仿射几何之类就差不多了。我刚开始是没有想到，因为我确实笨，也对非欧几何了解甚少。

我自大与否，很多人看得出来。我是爱写些半截答案和思路之类，以前没觉得不好，现在看来该改，因为影响别人思考了。但这也与自大无关，因为人家都能从字里行间知道那只是我的感想而已。

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关于种树问题答案的一个小注（图）

肯定还有共线的三点

WXC贴出此题了。还有两组共线的。

ever think about the reason of it?

你现在同意了？ [:B] 我已经老了。呵呵

还要保留9。

come on

回复：come on

你和 sean9991 中有一个是youhow 吧。

I dont think so

Simple? [:O]

of course it's very simple

回复：of course it's very simple

回复：回复：of course it's very simple

Sorry. I get your point.

[:%][:%]有人不厚道。