IMO 1959 Problem 5

An arbitrary point $M$ is selected in the interior of the segment $AB$ . The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$ , with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$ , intersect at $M$ and also at another point $N$ . Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$ .

a) Prove that $N$ and $N'$ coincide;

b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$ ;

c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$ .