# Thread: nine-sided regular polygon

1. ## nine-sided regular polygon

Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set { A1, A2, A3,.........A9 } ?

I think I am not understanding the problem correctly. I could only come up with 36, but the answer is 66.

Vicky

2. Originally Posted by Vicky1997
Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set { A1, A2, A3,.........A9 } ?

I think I am not understanding the problem correctly. I could only come up with 36, but the answer is 66.

Vicky
Hi Vicky,

the attachment shows 60 of the ways to place equilateral triangles
in the plane of the 9-gon, including at least 2 vertices of the n-gon.

In the first 3 sketches and some of the others,
just imagine moving the base of the triangle onto the next 9-gon side.

3. Originally Posted by Vicky1997
Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set { A1, A2, A3,.........A9 } ?

I think I am not understanding the problem correctly. I could only come up with 36, but the answer is 66.

Vicky
There are $\displaystyle \textstyle{9\choose2}=36$ ways of choosing two vertices of the nonagon. For each of those choices, there are two equilateral triangles having those two points as two of their vertices (the third vertex could be on either side of the line joining those two). That apparently gives 72 possible triangles. But there has been some double counting there. As shown in one of Archie Meade's graphics, it can happen that all three vertices of an equilateral triangle are vertices of the nonagon. There are three such triangles, and each of them has been counted three times. If you only count them once, that will reduce the total by 6, to 66 as claimed.

Originally Posted by Archie Meade
the attachment shows 60 of the ways to place equilateral triangles
in the plane of the 9-gon, including at least 2 vertices of the n-gon.
I think the question intends to say that at least two vertices of the triangle are vertices of the nonagon.

4. Thanks Opalg,

yes, the question is pointing to the vertices belonging to the triangle!
I didn't even consider it!!
I was 6 short of the total in any case!

5. Here is the corrected geometry, Vicky.
Opalg has demonstrated the math.

6. Thank You!!!

I definitely misunderstood the problem.