Question 83
Given a trapezium $ABCD$ ($AD \parallel BC$) with $AD = 3BC$. A circle $\Gamma_1$ with the center $B$ passes through the midpoint of $BD$, and a circle $\Gamma_2$ with the center $C$ passes through the midpoint of $AC$.
Prove that the line through the points of intersection of $\Gamma_1$ and $\Gamma_2$ meets the side $AD$ at its midpoint.
Prove that the line through the points of intersection of $\Gamma_1$ and $\Gamma_2$ meets the side $AD$ at its midpoint.