#### Question

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

#### Solution

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.

Is there an error in this question or solution?

Solution Two Circles Intersect in Points P and Q. a Secant Passing Through P Intersects the Circles in Aand B Respectively. Tangents to the Circles at a and B Intersect at T. A, Q, B and T Lie on a Circle. Concept: Cyclic Properties.