Question 7
Let $ABCD$ be a convex quadrilateral with the property that the circles having the segments $AB$ and $CD$ as diameters are tangent externally at a point $M$, different from the intersection point of the diagonals of the quadrilateral.
Let $K$ be the second point of intersection of the circumcircle of triangle $AMC$ with the line determined by $M$ and the midpoint of segment $AB$, and let $L$ be the second point of intersection of the circumcircle of triangle $BMD$ with the line determined by $M$ and the midpoint of segment $CD$.
Prove that $|MK - ML| = |AB - CD|$.
Let $K$ be the second point of intersection of the circumcircle of triangle $AMC$ with the line determined by $M$ and the midpoint of segment $AB$, and let $L$ be the second point of intersection of the circumcircle of triangle $BMD$ with the line determined by $M$ and the midpoint of segment $CD$.
Prove that $|MK - ML| = |AB - CD|$.