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

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Suppose CD and AB are two parallel tangents of a circle with center O

Construction: Draw a line parallel to CD passing through O i.e. OP

We know that the radius and tangent are perpendicular at their point of contact.

\(\angle\)OQC = \(\angle\)ORA = \(90^\circ\)

Now, \(\angle\)OQC + \(\angle\)POQ = \(180^\circ\)    (co-interior angles)

\(\Rightarrow\) \(\angle\)POQ = \(180^\circ\) - \(90^\circ\) = \(90^\circ\) 

Similarly, Now, \(\angle\)ORA + \(\angle\)POR = \(180^\circ\) (co-interior angles)

Hence, QR is a straight line passing through center O.

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