Question

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

Answer 1

et XBY and PCQ be two parallel tangents to a circle with centre O.
Construction: Join OB and OC.
Draw OA∥XY

Now, XB∥AO
⇒ ∠XBO + ∠AOB = 180°           (sum of adjacent interior angles is 180°)

Now, ∠XBO = 90°                      (A tangent to a circle is perpendicular to the radius through the point of contact)
⇒ 90° + ∠AOB = 180°
⇒ ∠AOB = 180° − 90° = 90°

Similarly, ∠AOC = 90°
∴ ∠AOB + ∠AOC = 90° + 90° = 180° 
Hence, BOC is a straight line passing through O.
Thus, the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.