Perpendicular Bisectors of a Triangle Meet

Donald has a triangle $ABC$ with side lengths $AB = 5$, $AC = 6$, and $BC = 7$. He intersects the perpendicular bisectors of $AB$ and $AC$ to get a point $P$, and he intersects the perpendicular bisectors of $AC$ and $BC$ to get a point $Q$. What is the distance between points $P$ and $Q$?