Question

Show that the line segment joining the points of contact of two parallel tangent passes through the centre.

Answer 1

Let $APB$ & $CQB$

be $2\parallel$ tangents

to circle, centre $O$.

To prove : $PQ$ proves through $O$

( Sum of interior ( alternate ) angles )

$\angle APO+\angle EOP={180}^o$

$\angle PO={90}^o$ $(\therefore$ tangent is $\perp$ to radius through print of contact)

$\angle EOP={180}^o-{90}^o$

$={90}^o$

Similarly, $\angle EOQ={90}^o$ ( Linear pair)

$\angle EOP+\angle EOQ={90}^o+906o={180}^o$

$\Rightarrow PQ$ is straight line passing through $O$.