[seqfan] Re: Number of Equilateral Triangles in Regular 3n-gon

Ignacio Larrosa Cañestro ilarrosa at mundo-r.com
Wed Oct 16 15:02:19 CEST 2013

```El 16/10/2013 9:23, Antreas Hatzipolakis escribió:
> A related problem:
>
> How many triangles can be formed from the vertices of a regular polygon
> with 13 sides so that the inside of each of the triangles contains the
> center of the circle circumscribing the polygon?
>
> Reference:
> http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=558360&
>
> Generalization for regullar 2n+1-gon.
>
> Antreas
>

You ask about the number of triangles acute, right A (n). The total
number of triangles is Comb (2n + 1, 3). As triangles there will be A
(n) = Comb (n, 3) - O (n) where O (n) is the number of obtuse-angled
triangles. Let us count O (n).

A_0 either one of 3n +1 vertices, which will locate the vertex
arbitrarily corresponding to the obtuse angle. The other two vertices
should be one on each side of the diameter passing through A_0. If one
of them is A_i, with i = 1 .. n, the other one may be one of the n - i
preceding A_0. Altogether

(n-1) + (n -2) + ... + (N - n) = 1 + 2 + ... + (N -1) = n (n-1) / 2

As A_0 can be any of the 2n +1 points,

O (n) = (2n +1) n (n-1) / 2

and

A (n) = Comb (n, 3) - O (n) = n (n + 1) (2n + 1) / 6 = Sum (k ^ 2, k, 1, n)

For a polygon of 13 sides, n = 6, A (n) = 91.

Attention: This is a translation of the Spanish automnática. Read it
with caution!

--
Saludos,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa at mundo-r.com
http://www.xente.mundo-r.com/ilarrosa/GeoGebra/

```