Squares on a Line Segment

point M

labels

A segment

AB

is given and on it the point

M

. On the same side of

AB

, the squares

AMKD

and

BMFE

are constructed. The circumcircles of the two squares, whose centers are

P

and

Q

, intersect at

M

and another point

N

.

(a) Prove that lines

FA

and

BK

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

.