Question 43
Let $ABCD$ be a cyclic quadrilateral with inradius $r$. Let $J$ and $K$ be the incenters of $ABC$ and $ACD$, and let $P$ and $Q$ be the circumcenters of $AJK$ and $CJK$, respectively. Prove that
$$
|PQ| = |AC| - \frac{S_{AJCK}}{r}.
$$
Here $S_{AJCK}$ denotes the area of the quadrilateral $AJCK$.
$$
|PQ| = |AC| - \frac{S_{AJCK}}{r}.
$$
Here $S_{AJCK}$ denotes the area of the quadrilateral $AJCK$.