KöMaL 2007 B4034

The midpoint of one side of a triangle is $F$, and the points dividing another side into three equal parts are $H_1$ and $H_2$. The third side is divided into $n$ equal parts by the points $N_1, N_2, \ldots, N_{n-1}$. Consider all triangles $FH_i N_j$ where $i$=1,2, $j=1,\ldots,n-1$. Show that for any triangle selected out of these triangles there is exactly one other triangle that has the same area.