Question 150
In the triangle $ABC$ $AB \gt AC$. A tangent to the circumcircle of the triangle $ABC$ has been drawn through the point $A$. This tangent intersects the line $BC$ at the point $P$. In continuation of the side $BA$ after the point $A$ point $Q$ was selected such that $AQ = AC$. Let $X$ and $Y$ be the midpoints of the segments $CQ$ and $AP$, respectively, and let $R$ belong to the segment $AP$ such that $AR = CP$. Prove that $CR = 2XY$.