Question 172
For an acute triangle $ABC$ satisfying $AB \neq AC$, denote by $H$ the foot of the perpendicular line segment drawn from the point $A$ to the side $BC$. Take points $P$ and $Q$ in such a way that the 3 points $A$, $B$, $P$ and the 3 points $A$, $C$, $Q$ lie on a straight line in the given order, respectively. If the 4 points $B$, $C$, $P$, $Q$ lie on the circumference of a circle, and if the condition $HP = HQ$ is satisfied, prove that the point $H$ must coincide with the circum-center of the triangle $APQ$.
Here we mean by $XY$ the length of the line segment $XY$.
Here we mean by $XY$ the length of the line segment $XY$.