Mathematics
Tangent at P to the circumcircle of triangle PQR is drawn. If this tangent is parallel to side, QR show that △PQR is isosceles.
Answer
Let DE be the tangent to the circle at P.
Given, DE || QR
∠EPR = ∠PRQ [Alternate angles are equal]
∠DPQ = ∠PQR [Alternate angles are equal] …….. (1)
As, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment, we have
Here, DE is tangent and PQ is chord.
∴ ∠DPQ = ∠PRQ ……… (2)
From (1) and (2),
⇒ ∠PQR = ∠PRQ
As, sides opposite to equal angles are equal.
⇒ PQ = PR.
Hence, proved that PQR is an isosceles triangle.
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