Question 159
Given a triangle $ABC$, not right-angled in $A$. Let $M$ be the midpoint of $BC$. Consider a point $D$ moving on the line $AM$, in such a way that $D$ is not coincident with $M$. Denote by $(O_1)$ the circle passing through $D$ and touching $BC$ in $B$; by $(O_2)$ the circle passing through $D$ and touching $BC$ in $C$. The line $AB$ intersects $(O_1)$ in the second point $P$ and the line $AC$ intersects $(O_2)$ in the second point $Q$. Show that the tangents at $P$ to $(O_1)$ and the tangents at $Q$ to $(O_2)$ intersect each other and the intersection point lies in a fixed line.