Prove that the segment, joining the points of contact of two parallel tangents passes through the centre.

Given: Two parallel tangents PQ and RS touch a circle C (O,r) at A and B respectively.
To prove: AB will pass through the centre O of the circle.
Construction:Draw a line OC parallel to RS.

Proof:               PA || CO

=> ∠PAO + ∠COA =180  [Sum of the angles on the same side of a transversal is 180]
=> 90 + ∠COA = 180 [ ∠PAO = Angle between a tangent and radius = 90]
=> ∠COA = 90

Similarly, ∠COB = 90
Therefore, ∠COA + ∠COB = 90 + 90 = 180

Hence, AOB is a straight line passing through O.