As a result of working on https://www.physicsforums.com/threads/area-of-hexagon-geometry-challenge.914759, this question occurred to me: Divide each side of a triangle into n equal lengths. Connect the ends of each length to the opposite vertex with straight lines, thereby forming 3n overlapping triangles of equal area. (The problem at the link is an example with n=3.) At how many points will three lines intersect? I can show this is equivalent to finding the number of integer solutions to abc = (n-a)(n-b)(n-c) This is easy for n even. For n odd I have not found any, but don't yet see how to prove there are none. [Edit: I mean it is easy to find some solutions for n even. If there are solutions for n odd then these lead to additional solutions for 2n.] A follow-on question: how many regions are formed? How many triangles, quadrilaterals, ..., hexagons?