Let $ABC$ be a triangle and let $X$ be a point in its interior. Point $S_A$ is the midpoint of arc $BC$ containing $X$ of the circumcircle of $BCX$. $S_B$ and $S_C$ are defined similarly. Prove that $S_A,S_B,S_C$ and $X$ are concyclic.
Source: Bulgarian Autumn Math Tournament 12.2
Tags: geometry, circumcircle
Let $ABC$ be a triangle and let $X$ be a point in its interior. Point $S_A$ is the midpoint of arc $BC$ containing $X$ of the circumcircle of $BCX$. $S_B$ and $S_C$ are defined similarly. Prove that $S_A,S_B,S_C$ and $X$ are concyclic.