Question 40
We consider two circles in the plane, which intersect at two different points $X$ and $Y$.
Prove that there are four fixed points in this plane with the following property: For every circle that lies at the intersection of the two given circles and touches them at the points $A$ and $B$ and intersects the line $X Y$ at the points $C$ and $D$, each of the lines $A C, A D, B C$ and $B D$ through one of these four points.
Prove that there are four fixed points in this plane with the following property: For every circle that lies at the intersection of the two given circles and touches them at the points $A$ and $B$ and intersects the line $X Y$ at the points $C$ and $D$, each of the lines $A C, A D, B C$ and $B D$ through one of these four points.