Problem 4

Let

C

be a circle of radius

R

and center

O

, and let

S

be a fixed interior point. Let

AA'

and

BB'

be perpendicular chords through

S

. The rectangles

SAMB

SB'N'A'

SA'M'B'

, and

SBNA

are formed. Determine the locus of the four points

M,N',M',N

A

moves around

C

Step 1 of 5: Combine the four small rectangles

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MN'M'N\text{ is a rectangle with }MN'\parallel AA',\quad NM'\parallel BB'

Detailed analysis

From the four rectangle constructions, $MN'M'N$ is a rectangle whose sides are parallel to $AA'$ and $BB'$ . The perpendicular bisectors of $AA'$ and $BB'$ pass through $O$ ; the same lines are the perpendicular bisectors of $MN'$ and $NM'$ . Hence $O$ is the center of $MN'M'N$ .