Question 5
(ROM) A segment $AB$ is given and on it a point $M$. On the same side of $AB$ squares $AMCD$ and $BMFE$ are constructed. The circumcircles of the two squares, whose centers are $P$ and $Q$, intersect in $M$ and another point $N$.
(a) Prove that lines $FA$ and $BC$ intersect at $N$.
(b) Prove that all such constructed lines $MN$ pass through the same point $S$, regardless of the selection of $M$.
(c) Find the locus of the midpoints of all segments $PQ$, as $M$ varies along the segment $AB$.
(a) Prove that lines $FA$ and $BC$ intersect at $N$.
(b) Prove that all such constructed lines $MN$ pass through the same point $S$, regardless of the selection of $M$.
(c) Find the locus of the midpoints of all segments $PQ$, as $M$ varies along the segment $AB$.