Triangles and Circles But Not Geometry Part -5

A right triangle inscribed in a circle must have as its hypotenuse a diameter of the circle. (This is because, for any right triangle, the midpoint of the hypotenuse is equidistant from each of the vertices, and hence is the center of the circle in which the triangle is inscribed.)

Now imagine a $2n$ sided regular polygon inscribed in a circle. Then there are $n$ diameters of the circle that can be formed by connecting vertices of the polygon. For each of these diameters, we can form $2n - 2$ right triangles by connecting the endpoints of a diameter, in turn, to each of the $2n - 2$ remaining vertices of the polygon.

Thus in general, for a $2n$ sided regular polygon, we can form $n(2n - 2)$ right triangles by joining vertices of the polygon. For $n = 96$ we can thus form $96*(2*96 - 2) = \boxed{18240}$ right triangles.