Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Let lines AP and BR are parallel tangents to circle having centre O

We have to prove that AB is the diameter

To prove AB as diameter, we have to prove that AB passes through O which means that points A, O and B are on the same line or collinear

OA is perpendicular to PA at A because the line from the centre is perpendicular to the tangent at the point of contact

PA || RB

Hence OA is also perpendicular to RB

⇒ OA perpendicular to PA and RB …(i)

Similarly, OB is perpendicular to RB at B because the line from the centre is perpendicular to the tangent at the point of contact

PA || RB

Hence OB is also perpendicular to PA

⇒ OB perpendicular to PA and RB …(ii)

From (i) and (ii) we can say that OA and OB can be same line or parallel lines, but we have a common point O which implies that OA and OB are same lines

Hence A, O, B lies on the same line, i.e. A, O and B are collinear

Thus AB passes through O

Hence AB is the diameter

Hence, the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.