Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangent at P and Q intersect at point T; show that the points P, B, Q and T are concyclic.

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Solution

Let $∠ T P Q = θ$

Therefore $∠ T Q P = θ$

Also PT is tangent to the circle therefore, the angle between the the tangent PT and chord PA is equal to the the angle made by the chord on the opposite segment ( $∠ P B A = θ$ ) similarly $∠ Q B A = ∠ T Q P = θ$

Also $∠ P T Q = (180 - (2 \times θ$ )) by angle sum property of triangle.

In quadrilateral TQBP, angle T and B add upto 180 ,therefore TQBP is a cyclic quadrilateral

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