IrMO 2016 Problem 10

Let $AE$ be a diameter of the circumcircle of triangle $ABC$ . Join $E$ to the orthocentre, $H$ , of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$ . Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$ .

Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.