Question 116
Given a circle and four points $B$, $C$, $X$, $Y$ on it, let $A$ be the midpoint of segment $BC$ and $Z$ the midpoint of segment $XY$. Through $B$ and $C$ draw the lines $L_1$ and $L_2$ perpendicular to $BC$, respectively. Let the line through $X$ perpendicular to $AX$ meet $L_1$, $L_2$ at the points $X_1$, $X_2$, respectively, and let the line through $Y$ perpendicular to $AY$ meet $L_1$, $L_2$ at the points $Y_1$, $Y_2$, respectively. Let $X_1Y_2$ and $X_2Y_1$ intersect at the point $P$. Prove that $\angle AZP = 90^\circ$.