Question 75
A cyclic hexagon $ABCDEF$ is given with the following property:
$$
AB \cdot CD \cdot EF = BC \cdot DE \cdot AF.
$$
Let $B_1$ be the symmetric point of $B$ with respect to line $AC$, $D_1$ be the symmetric point of $D$ with respect to line $CE$, and $F_1$ be the symmetric point of $F$ with respect to line $EA$. Prove that $\triangle B_1D_1F_1 \sim \triangle BDF$.
$$
AB \cdot CD \cdot EF = BC \cdot DE \cdot AF.
$$
Let $B_1$ be the symmetric point of $B$ with respect to line $AC$, $D_1$ be the symmetric point of $D$ with respect to line $CE$, and $F_1$ be the symmetric point of $F$ with respect to line $EA$. Prove that $\triangle B_1D_1F_1 \sim \triangle BDF$.