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 4 of 5: Convert the constant sum into a fixed radius

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MN'^2+M'N^2=MM'^2=4OM^2

Detailed analysis

Since $MN'M'N$ is a rectangle, its diagonals are equal and perpendicular, so the sum of the squares of two adjacent sides is $MM'^2$ . Since $O$ is its center, $MM'=2OM$ , hence $MN'^2+M'N^2=4OM^2$ . Comparing with the previous step shows $OM$ is constant.