let $pq$ be the diameter of a circle. let $a$ and $b$ be points on the circle, and let the tangents to the circle through $a$ and $b$ intersect at $c$. let the tangent to the circle at $q$ intersect $pa$, $pb$, and $pc$ at $a'$, $b'$, and $c'$, respectively. show that $c'$ is the midpoint of $a'b'$.
