NL-NWO 1996 Problem 4

A line $l$ intersects the segment $AB$ perpendicular to $C$ . Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$ . The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$ . a. Prove that quadrilateral $CFDE$ is a rectangle. b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$ .