Question 38
Let $\Gamma$ be a circle and $AB$ be a diameter. Let $\ell$ be a line outside the circle, and is perpendicular to $AB$. Let $X, Y$ be two points on $\ell$. If $X'$ and $Y'$ are two points on $\ell$ such that $AX$ and $BX'$ intersect on $\Gamma$ and such that $AY$ and $BY'$ intersect on $\Gamma$, prove that the circumcircles of the triangles $AXY$ and $AX'Y'$ intersect at a point on $\Gamma$ other than $A$, or the three circles are tangent at $A$.