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.
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.