Equal-area triangulations
Source: IMO 2015 Shortlist, G8
July 7, 2016
geometrytriangulationIMO Shortlistpolygon
Problem Statement
A triangulation of a convex polygon is a partitioning of into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thaiangulation if all triangles in it have the same area.Prove that any two different Thaiangulations of a convex polygon differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)Proposed by Bulgaria