Question 119
In triangle $ABC$ we have $|AC| \neq |BC|$. Let $S$ be the midpoint of the side $AB$, $D$ the intersection of the side $AB$ with the bisector of the angle $\angle ACB$, and $E$ the mirror image of the point $D$ under reflection in the point $S$. The circumcircle of triangle $ABC$ meets the line $CD$ at the points $C$ and $F$, and the line $CS$ at the points $C$ and $G$. Prove that the points $E$, $G$, $F$ and $S$ lie on one circle.