Question 24
(USA 5) A circle $\mathcal O$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB\times CQ=(BC/2)^2$, prove that line segment $PQ$ is tangent to circle $\mathcal O$, and prove the converse.