Question 81
Let $\omega$ be a circle and $A$ a point outside of $\omega$. Draw the tangents from $A$ to $\omega$ and call the points of tangency $X$ and $Y$. Let $B$ and $C$ be points on the segments $AX$ and $AY$, respectively, such that the perimeter of $\triangle ABC$ is equal to the length of the segment $AX$. Let $D$ be the reflection of $A$ in the line $BC$. Show that the circumcircle $BDC$ touches $\omega$.