19-gon and Pigeonhole principle

1. Dec 16, 2012

Numeriprimi

Hey!
I have got some question for you.

Decide if you can choose seven tops of the regular 19-gon and four of them are tops of trapezoid.
(I think - Pigeonhole principle, but how?)

2. Dec 17, 2012

tiny-tim

Hi Numeriprimi!

What do you mean by "tops"?

3. Dec 17, 2012

I think he means the vertices. How I would try to prove it. Feel free to stop reading once you think that you have the answer...

1) A trapezoid is defined by having two parallel sides. So you want to construct a set of points and none of the connections between the points are to be parallel.

2) If we numerate the points we can start forming all the families of parallel lines.

3) If we enumerate in a circle one family is {(2,19),(3,18),(4,17),(5,16),..., (10,11)} You see that even one "length 1" pair is included.

4) There are 19 of these families, and they account for all the possible connections there are.

5) The possible connections between n points are (n^2-n)/2

6) Pidgeonhole

4. Dec 17, 2012

Numeriprimi

Yes, I mean vertices... Sorry for my English because is quite hard to choose right word with same meaning in my language when is a lot of words :-)

So, I will read and understand your answer after school because I going to sleep. Then I will write when I won't understand you.

For now... thanks very much :-)

5. Jan 2, 2013