ÖMO 2006 Problem 3

The triangle $ABC$ is given. On the extension of the side $AB$ we construct the point $R$ with $BR = BC$ , where $AR > BR$ and on the extension of the side $AC$ we construct the point $S$ with $CS = CB$ , where $AS > CS$ . Let $A_1$ be the point of intersection of the diagonals of the quadrilateral $BRSC$ . Analogous we construct the point $T$ on the extension of the side $BC$ , where $CT = CA$ and $BT > CT$ and on the extension of the side $BA$ we construct the point $U$ with $AU = AC$ , where $BU > AU$ . Let $B_1$ be the point of intersection of the diagonals of the quadrilateral $CTUA$ . Likewise we construct the point $V$ on the extension of the side $CA$ , where $AV = AB$ and $CV > AV$ and on the extension of the side $CB$ we construct the point $W$ with $BW = BA$ and $CW > BW$ . Let $C_1$ be the point of intersection of the diagonals of the quadrilateral $AVWB$ . Show that the area of the hexagon $AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ABC$ and $A_1B_1C_1$ .