A straight line is drawn cutting two equal circles and passing through the mid-point m of the line joining the centre o and o'. Prove that the chord AB and CD intersect by 2 circles are equal

The figure is given below:

Draw OP⊥AB and O'Q⊥CD.

Consider ΔOPM and ΔO'QM

OM = O'M [Given]

∠OMP = ∠O'QM [Vertically opposite angles]

∠OPM = ∠O'QM = 90° [Construction]

∠POM = ∠QO'M [Angle sum property of a triangle]

So, ΔOPM ≅ ΔO'QM

OP = O'Q [CPCT]

⇒ AB = CD [Equal chords of same circle or equal circles are equidistance from the centres of the respective circles.]

Hence proved.