Let P be a convex polygon and <spanclass=′latex−bold′>T</span> be a triangle with vertices among the vertices of P. By removing <spanclass=′latex−bold′>T</span> from P, we end up with 0,1,2, or 3 smaller polygons (possibly with shared vertices) which we call the effect of <spanclass=′latex−bold′>T</span>. A triangulation of P is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of P beautiful, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of P is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.
combinatoricstriangulationconvex quadrilateralcombinatorical geometry