Question

Tangent at P to the circumcircle of triangle PQR is drawn. If the tangent is parallel to side, QR show that ΔPQR is isosceles.

Answer 1

DE is the tangent to the circle at P.

DE || QR  ...(Given)

∠EPR = ∠PRQ   ...(Alternate angles are equal)

∠DPQ = ∠PQR  (Alternate angles are equal) ...(i)

Let ∠DPQ = x and ∠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

∴ ∠DPQ = ∠PRQ  ...(ii) (DE is tangent and PQ is chord)

From (i) and (ii)

∠PQR = ∠PRQ

`=>` PQ = PR

Hence, triangle PQR is an isosceles triangle.