Question 92
Let $k$ be a circle and $AB$ a chord of $k$ such that the centre of $k$ does not lie on $AB$. Let $C$ be a point on $k$ different from $A$ and $B$. For each choice of $C$, let $P_{C}$ and $Q_{C}$ be the projections of $A$ onto $BC$ and of $B$ onto $AC$, respectively. Further, let $O_{C}$ be the circumcentre of the triangle $P_{C} Q_{C} C$. Show that there is a circle $\omega$ such that $O_{C}$ lies on $\omega$ for every choice of $C$.