Question 137
Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $N$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centres of the circles $AMN$ and $AXY$ passes through a fixed point.