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 2 of 5: Relate the two perpendicular chords to the radius

Previous step Next step

AB^2+A'B'^2=4R^2

Detailed analysis

Let $D$ be the second intersection of $MN'$ with $C$ . Equal inscribed angles give $AD=AB$ . Also $DA'$ is parallel to $MN'$ , so the angle $DA'B'$ is right; therefore $D$ and $B'$ are diametrically opposite. Thus $A'D^2+D B'^2=4R^2$ , and substituting $AD=AB$ yields $AB^2+A'B'^2=4R^2$ .