Question

Tangent at $P$ to the circumcircle of triangle $PQR$ is drawn. If this tangent is parallel to side $QR$ show that $△PQR$ is an isosceles triangle.

Answer 1

DE is the tangent to the circle at P.

$DE||QR$ [Given]

$\angle EPR=\angle PRQ$ [Alternate angles are equal]

$\angle DPQ=\angle PQR.....\left(i\right)$ [Alternate angles are equal]

Let $\angle DPQ=x$ and $\angle EPR=y$

Since the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment

$\therefore \angle DPQ=\angle PRQ....\left(ii\right)$ [DE is tangent and PQ is chord]

From (i) and (ii)

$\angle PQR=\angle PRQ$

$\therefore PQ=PR$

Hence, $△PQR$ is an isosceles triangle.