TUYMAADA 2005 Problem 7

Let $I$ be the incentre of triangle $ABC$ . A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$ . Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$ .

Proposed by S. Berlov