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.

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