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 4 of 5: Find the fixed point on the arc

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\text{The bisector of the fixed arc }AB\text{ is a fixed point }S\in\odot(AB)

Detailed analysis

On the fixed circle with diameter $AB$ , the internal bisector of $\angle ANB$ passes through the midpoint $S$ of the fixed arc $AB$ on the chosen side. Since $MN$ is that bisector, every line $MN$ passes through $S$ , independent of $M$ .