#### 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

#### Solution

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, Δ PTQ,

`∠`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.

Is there an error in this question or solution?

Solution 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 Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.