If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 10.25).

Draw a perpendicular from the center of the circle ON to the line AD.

We can see that BC is the chord of the smaller circle, and AD is the chord of the bigger circle.

We know that perpendicular drawn from the center of the circle bisects the chord.

∴ BN = NC ... (1)

and, AN = ND ... (2)

Subtracting (2) from (1), we obtain

AN − BN = DN − CN

∴ AB = CD