AoPS - Problem

Let

$ABC$ be an acute scalene triangle. Let

$D$ be the foot of the

$A$ -bisectrix and

$E$ be the foot of the

$A$ -altitude. The perpendicular bisector of the segment

$AD$ intersects the semicircles of diameter

$AB$ and

$AC$ which lie on the outside of triangle

$ABC$ at

$X$ and

$Y$ respectively. Prove that the points

$X,Y,D$ and

$E$ lie on a circle.