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.

Answer 1

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.