Question 192
Consider triangle $ABC$ with $|AB| \lt |AC| \lt |BC|$, and let $I$ be its incentre. The incircle of $\triangle ABC$ touches sides $BC$ and $AC$ at points $D$ and $E$, respectively. Let $R$ denote the midpoint of side $AC$. The point $N$ lies on the segment $DE$ such that $|RE| = |RN|$. Let $Q$ be the midpoint of segment $ND$. The lines $AN$ and $RQ$ intersect at point $P$. Prove that the points $I, N, P, Q$ lie on the same circle.