Question 72
Let $I$ be the incenter of a triangle $ABC$. The circle passing through the vertex $B$ and touches the line $AI$ at $I$ intersects the sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Let $R$ be the intersection point of the line $QI$ and the side $AC$. Prove that
$$
|AR| \cdot |BQ| = |PI|^2.
$$
$$
|AR| \cdot |BQ| = |PI|^2.
$$