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
Join AB, PB and BQ
TP is the tangent and PA is a chord
∴ `∠`TPA = `∠`ABP …… (i) (angles in alternate segment)
`∠`TQA = `∠`ABQ ……. (ii)
Adding (i) and (ii)
`∠`TPA+ `∠`TQA = `∠`ABP + `∠`ABQ
But, Δ PTQ,
`∠`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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