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 3 of 5: Compute the constant sum of the two side squares

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MN'^2+M'N^2=AB^2+A'B'^2+4SA\cdot SA'=8R^2-4OS^2

Detailed analysis

Expanding the squared side lengths of the two rectangles and using Pythagoras in the right triangles at $S$ gives $MN'^2+M'N^2=AB^2+A'B'^2+4SA\cdot SA'$ . The preceding step turns this into $8R^2-4OS^2$ , which is independent of the position of $A$ .