Question 115
Let $ABC$ be a triangle that satisfies $AB = 13$, $BC = 14$, $AC = 15$. Given a point $P$ in the plane, let $P_A$, $P_B$, $P_C$ be the reflections of $A$, $B$, $C$ across $P$. Call $P$ good if the circumcircle of $P_A P_B P_C$ intersects the circumcircle of $ABC$ at exactly 1 point. The locus of good points $P$ encloses a region $\mathcal{S}$. Find the area of $\mathcal{S}$.