Problem:
A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
Solution:
There are equilateral and other isosceles triangles on each base, providing a total of triangles when all vertices are chosen from the same base.
To count the rest of the isosceles triangles, there are ways to choose the base that will contain vertices. Assume that vertices have been chosen from the bottom base. Then there are no isosceles triangles if the vertices on the bottom base are adjacent vertices of the hexagon, isosceles triangles if the vertices on the bottom base have vertex between them on the hexagon, and isosceles triangles if the vertices on the bottom base have vertices between them on the hexagon.
The total number of isosceles triangles is .
The problems on this page are the property of the MAA's American Mathematics Competitions