Question 150
Let $ABC$ be a triangle. Point $D$ lies on segment $BC$ such that $\angle BAD = \angle DAC$. Point $X$ lies on the opposite side of line $BC$ as $A$ and satisfies $XB = XD$ and $\angle BXD = \angle ACB$. Analogously, point $Y$ lies on the opposite side of line $BC$ as $A$ and satisfies $YC = YD$ and $\angle CYD = \angle ABC$. Prove that lines $XY$ and $AD$ are perpendicular.