, 2010

Any triangle whose vertices are among the vertices of a regular hexagon is one of the following:
* a triangle $\Delta_1$ whose vertices are three consecutive vertices of the hexagon;
* a triangle $\Delta_2$ whose two vertices are adjacent vertices of the hexagon and the third one is adjacent to none of the first two;
* a triangle $\Delta_3$ where any two vertices are not adjacent vertices of the hexagon.

Since the areas of the chosen triangles are different, the triangles must be equal to the triangles $\Delta_1$, $\Delta_2$, $\Delta_3$. The hexagon can be divided into four parts (Fig. 9): the triangle $\Delta_3$ surrounded by three triangles $\Delta_1$. The area of the triangle $\Delta_2$ (marked by a dotted line in Fig. 9) is twice the area of the triangle $\Delta_1$ because they have the same base but the height of $\Delta_2$ is twice the height of $\Delta_1$.