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 figure).

Let OP be the perpendicular from O on linel . Since, the perpendicular from the centre of a circle to a chord bisects the chords.

Now, BC is the chord of the smaller circle and $$OP\perp{BC}$$.

Therefore, BC = PC ...(i)
Since, AD is a chord of the larger circle and $$OP\perp{AD}$$.

Therefore, AP = PD ...(ii)

On subtracting Eq. (i) from Eq.(ii), we get,
AP - BP = PD - PC

Thus, AB = CD
Hence, it is proved.