Problem 5

An arbitrary point

M

is selected in the interior of segment

AB

. Squares

AMCD

and

MBEF

are constructed on the same side of

AB

, with circumcenters

P

and

Q

. Their circumcircles meet again at

N

. Let

N'

be the intersection of

AF

and

BC

. (a) Prove

N=N'

. (b) Prove that

MN

passes through a fixed point independent of

M

. (c) Find the locus of the midpoint of

PQ

M

varies.

Step 5 of 5: Compute the locus of the square-center midpoint

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R=\operatorname{mid}(PQ)\quad\Longrightarrow\quad d(R,AB)=\frac{AB}{4}

Detailed analysis

The centers $P$ and $Q$ lie at distances $AM/2$ and $MB/2$ from $AB$ , respectively, on the same side. Thus the midpoint $R$ has distance $(AM/2+MB/2)/2=AB/4$ from $AB$ . As $M$ varies, its projection along $AB$ varies over the corresponding interval, so the locus is the segment on the line parallel to $AB$ at distance $AB/4$ (between the endpoint positions obtained as $M$ approaches $A$ and $B$ ).