2015 IMO Shortlist G8

A triangulation of a convex polygon $\Pi$ is a partitioning of $\Pi$ 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 $\Pi$ 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.) (Bulgaria)

凸多边形 $\Pi$ 的三角剖分是通过对角线将 $\Pi$ 分割成三角形，除了多边形的顶点之外，没有公共点。如果三角剖分中的所有三角形都具有相同的面积，我们就说三角剖分是泰式剖分。证明凸多边形 $\Pi$ 的任何两个不同的泰角投影恰好相差两个三角形。 (换句话说，证明可以用一对不同的三角形替换第一个泰式测量中的一对三角形，从而获得第二个泰式测量。)(保加利亚)