Show that the line segment joining the points of contact of two parallel tangents passes through the centre.

Let AB and CD are parallel tangents of circle with centre O.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

∠APO = 90°

⇒ ∠APO + ∠EOP = 180° (sum of adjacent interior angles)

⇒ ∠EOP = 180° - 90°

⇒ ∠EOP = 90°

Similarly,

⇒ ∠EOQ + ∠CQO = 180° (sum of adjacent interior angles)

⇒ ∠EOQ = 180° - 90°

⇒ ∠EOQ = 90°

∠EOP + ∠EOQ = 90° + 90° = 180°

∴ POQ is a straight line.

Hence, line segment joining the points of contact of two parallel tangents passes through the centre.