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Question
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.
Solution
Join AB, PB and BQ
TP is the tangent and PA is a chord
∴ ∠TPA = ∠ABP ...(i) (Angles in alternate segment)
Similarly,
∠TQA = ∠ABQ ...(ii)
Adding (i) and (ii)
∠TPA + ∠TQA = ∠ABP + ∠ABQ
But, ΔPTQ,
∠TPA + ∠TQA + ∠PTQ = 180°
`=>` ∠PBQ = 180° – ∠PTQ
`=>` ∠PBQ + ∠PTQ = 180°
But they are the opposite angles of the quadrilateral
Therefore, PBQT are cyclic.
Hence, P, B, Q and T are concyclic.
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