Mathematics
Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; show that the points P, B, Q and T are concyclic.
Circles
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Answer
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 :
From figure,
TP is a tangent and PA is a chord.
∴ ∠TPA = ∠ABP ……….(1)
Also,
TQ is a tangent and AQ is a chord.
∴ ∠TQA = ∠ABQ ……….(2)
Adding (1) and (2), we get :
∠TPA + ∠TQA = ∠ABP + ∠ABQ = ∠PBQ
In △PTQ,
⇒ ∠TPA + ∠TQA + ∠PTQ = 180°
⇒ ∠PBQ + ∠PTQ = 180°.
∠PBQ and ∠PTQ are opposite angles of a quadrilateral.
Hence, proved that P, B, Q and T are concyclic.
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