Question 153
As shown in the figure, in an acute triangle $ABC$ with $AB \lt AC$, let $AH$ be its altitude and $G$ the barycentre. Let $P, Q$ be the tangent points of the incircle to $AB, AC$, respectively. Let $M, N$ be the midpoint of $BP, CQ$, respectively. Let $D, E$ be two points lying on the incircle of the triangle $ABC$ such that
$$
\angle BDH + \angle ABC = 180^\circ, \quad \angle CEH + \angle ACB = 180^\circ.
$$
Prove that the lines $MD, NE, GH$ are concurrent.
$$
\angle BDH + \angle ABC = 180^\circ, \quad \angle CEH + \angle ACB = 180^\circ.
$$
Prove that the lines $MD, NE, GH$ are concurrent.