Two circles that have the same center point are called concentric circles. A secant line that intersects both of the concentric circles creates two congruent segments between the two circles.

## Problem

Two congruent circle with center at point O are intersected by a secant. Prove that AB=CD.

## Strategy

The secant creates two chords- AD in the outer circle, and BC in the inner circle. What do we know about chords? One property is that a diameter or radius that is perpendicular to a chord bisects that chord.

If we draw such a radius, it bisects both AD and BC – so BE=EC and AE=ED. And with some simple math, we can show that the non-overlapping difference between the line segments is also equal.

## Proof

(1) OE⊥BC //Construction

(2) BE=EC //Perpendicular radius bisects the chord

(3) AE=ED //Perpendicular radius bisects the chord

(4)AE-BE=ED-EC //(2), (3) , Subtraction property of equality

(5)AB=CD //Segment addition postulate