Question 31
Let $ABC$ be a triangle. At the extensions of the side $AB$ we consider the points $A_1$, $B_1$ such that $AB_1 = BA_1 = x$ ($B_1$ closer to $A$). At the extensions of the side $BC$ we consider the points $B_4$, $C_4$ such that $CB_4 = BC_4 = y$ ($B_4$ closer to $C$). At the extensions of the side $AC$ we consider the points $C_1$, $A_4$ such that $AC_1 = CA_4 = z$ ($A_4$ closer to $C$). At the interior of the segment $A_1A_4$, we consider points $A_2$, $A_3$ such that $A_1A_2 = A_3A_4 = mA_1A_4$ (where $m \in \mathbb{R}$ with $0 \lt m \lt \frac{1}{2}$). Similarly we define the points $B_2$, $B_3$ and $C_2$, $C_3$ at the interior of the segments $B_1B_4$ and $C_1C_4$, respectively. If $D = BB_2 \cap CC_2$, $E = AA_3 \cap CC_2$, $F = AA_3 \cap BB_3$, $G = BB_3 \cap CC_3$, $H = AA_2 \cap CC_3$ and $I = AA_2 \cap BB_2$, prove that the diagonals $DG$, $EH$, $FI$ of the hexagon $DEFGHI$ are concurrent.
