In the image below, $A(10,10)$ is the center of a circle where $B(6,7),$ $C(13,6)$ and $Q$ lie on the circumference.

Given that the angles $\mathrm{\angle }CAP$ and $\mathrm{\angle }PAB$ are equal, find length of the line segment $PQ$.

$AB$, $AC$ and $AQ$ are each a radius of the same circle.

$P$ is the midpoint of $BC.$

$|PQ|=|AQ|-|AP|$ so you don't need the coordinates of $Q$