Question

Two circles intersect each other at P and Q. A is any point on the line PQ; AB and AC are tangents from A to the circles. Prove that AB=AC​

Answer 1

Join PQ,AQ and QB

TA is a tangent and AP is a chord

∴  ∠TAP=∠PQA        --- ( 1 )  [ Angles in alternate segment ]

Similarly, 

⇒  ∠TBP=∠PQB        ---- ( 2 )

Adding ( 1 ) and ( 2 ),

⇒  ∠TAP+∠TBP=∠PQA+∠PQB

But △ATB,

⇒  ∠TAP+∠TBP+∠ATB=180o

⇒  ∠AQB=180o−∠ATB

⇒  ∠AQB+∠ATB=180o

But they are the opposite angles of the quadrilateral.

∴   AQBT are cyclic.

Hence, A,Q,B and T are concyclic