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 5 of 5: State the locus

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M,N',M',N\in\{X:OX=OM\}

Detailed analysis

The four vertices of the rectangle $MN'M'N$ are all at distance $OM$ from its center $O$ . As $A$ traverses the whole circle $C$ , the perpendicular chord $AA'$ takes every direction, so the rectangle rotates through all directions; hence each of the four vertices traverses the entire fixed circle centered at $O$ with radius $OM$ .