Question 122
Given a trapezium $ABCD$ with $BC \parallel AD$, $AD \gt BC$. Let $M$ be the point of intersection of $AC$ and $BD$. A circle $\Gamma_1$ passes through $M$ and touches $AD$ at $A$. A circle $\Gamma_2$ passes through $M$ and touches $AD$ at $D$. Let $S$ be the point of intersection of the lines $AB$ and $DC$, $X$ be the point of intersection of the circle $\Gamma_1$ and the line $AS$, $Y$ be the point of intersection of the circle $\Gamma_2$ and the line $DS$, $O$ be the circumcenter of the triangle $ASD$.
Prove that $SO \perp XY$.
Prove that $SO \perp XY$.