Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Take two congruent circles, and let their centres be O1 and O2. Let the chords making equal angles at centres be AB and CD respectively for two circles. Let the radius of both circles be r. We know that O1A = O1B = O2C = O2D. Since O1A = O1B = O2C = O2D, we get ∠O1AB = ∠O1BA and ∠O2CD = ∠O2DC. So, we get ∠AO1B = ∠CO2D. So, by SAS congruency, we get triangle AO1B is congruent to triangle CO2D. ∴AB = CD, which implies that the length of chords are equal.