Question 93
Given an acute triangle $ABC$ let $D$ denote the foot of the altitude from $A$, and let $H$ be the orthocentre. Drop the tangent from the point $B$ to the circle centered at $H$ with radius $HD$ and let $P$, distinct from $D$, be the point where the tangent touches the circle. Drop the tangent from the point $C$ to the circle centered at $A$ with radius $AD$ and let $R$, distinct from $D$, be the point where the tangent touches the circle. Prove that the line $PR$ passes through two of the feet of the altitudes of the triangle $ABC$. (Inspired by a problem from Estonian Selection Exams 2015.)