2001 OIM Problems/Problem 2

Problem

The circle inscribed in the triangle $ABC$ has center $O$ and is tangent to the sides $BC$ , $AC$ and $AB$ at points $X$ , $Y$ and $Z$ , respectively. The lines $BO$ and $CO$ intersect to the line $YZ$ at points $P$ and $Q$ , respectively. Show that if the segments $XP$ and $XQ$ have the same length, then the triangle $ABC$ is isosceles.