Prove that the intercept of a tangent between two parallel tangents to a circle subtends right angle at the centre.

Let $XY \parallel X'Y'$ be two parallel tangents with $LM$ as diameter. $\Delta OAL \cong \Delta OAC \implies \angle 1 = \angle 2$. Similarly, $\angle 3 = \angle 4$. But $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 180^\circ \implies 2 \angle 1 + 2 \angle 3 = 180^\circ$ or $\angle 1 + \angle 3 = 90^\circ \implies AB$ subtends right angle at the centre.