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 3 of 5: Relate the variable point to

N

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\frac{AM}{MB}=\frac{CM}{MB}=\frac{AN}{NB}

Detailed analysis

Triangles $ABN$ and $BCM$ are similar: they have the angle at $B$ in common and each has a right angle. Therefore $AM/MB=CM/MB=AN/NB$ , so $MN$ bisects $\angle ANB$ .