Question

Two circles intersect in points P and Q. A secant passing through P intersects the circles in A and B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, B and T lie on a circle.

Answer 1

Join PQ.

AT is tangent and AP is a chord.

∴ ∠TAP = ∠AQP  (Angles in alternate segments) ...(i)

Similarly, ∠TBP = ∠BQP ...(ii)

Adding (i) and (ii)

∠TAP + ∠TBP = ∠AQP + ∠BQP

`=>` ∠TAP + ∠TBP = ∠AQB  ...(iii)

Now in ΔTAB,

∠ATB + ∠TAP + ∠TBP = 180°

`=>` ∠ATB + ∠AQB = 180°

Therefore, AQBT is a cyclic quadrilateral.

Hence, A, Q, B and T lie on a circle.