Question 22
Let $ABC$ be a triangle right-angled at $A$. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ at $M$, respectively $N$. Prove that if $BM \cdot CN \cdot BC = MN^3$, then the symmetric point of $A$ with respect to the midpoint of the segment $MN$ belongs to $BC$.