Question 15
Let $\Gamma_1,\Gamma_2$ be intersecting circles with centers $O_1,O_2$, respectively, such that $\Gamma_2$ meets the segment $O_1O_2$ at $A$. Their intersection points are $C,D$. The line $AD$ meets $\Gamma_1$ again at $S$. The line $CS$ meets $O_1O_2$ at $F$. Let $\Gamma_3$ be the circumcircle of triangle $ADF$, and let $E$ be the second intersection of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.
