Determining the number of triangles in a 15-gon!

Note that a triangle is obtuse if and only if the center of the circle is not in the interior of the triangle (consider the arc intercepted by the largest angle). Now, consider two vertices that do not lie on the same diameter (call them A and B) and the diameters passing through them:

There must be at least at least one vertex in the red region, otherwise the 15-gon would not contain the center of the circle; call this point C. Consider the following important facts:

1) Three points in the red region will never form an acute triangle.

2) Two points in the red region, along with A or B, will never form an acute triangle.

3) A point in the blue region and a point in the red region will form an acute triangle with either A or B (but not both).

4) A point in the red region, along with A and B, will always form an acute triangle.

5) Three points in the blue region (which could be A or B), may or may not form an acute triangle.

Note that this exhausts all possibilities. Thus, the number of acute triangles formed will be $a=0 + 0 + rb + r + X$ , where $r$ and $b$ represent the number of points (discounting A and B) in the red and blue regions, respectively, and $X$ is the number of acute triangles formed among the points in the blue region. The terms are ordered such that each one is the number of acute triangles added by each fact listed above. Our goal is to minimize $a$ . Since C must be in the red region, $r\geq 1$ . If $b=0$ , $a = r$ . If $b>0$ , $a = rb + r + X > r$ . Thus $a$ is minimized when $b = 0$ , and the number of obtuse triangles is $T - r$ , where $T$ is the total number of triangles formed by vertices of the 15-gon, which is ${15\choose 3}$ . Since every vertex that is not A or B must be in the red region, $r=13$ . So the number of obtuse triangles is ${15\choose 3}-13 = 442$