Question

Two circles intersect at 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, T and B are concyclic.

Answer 1

Join $PQ,AQ$ and $QB$

$TA$ is a tangent and $AP$ is a chord

$\therefore$ $\angle TAP=\angle PQA$ --- ( 1 ) [ Angles in alternate segment ]

Similarly,

$\Rightarrow$ $\angle TBP=\angle PQB$ ---- ( 2 )

Adding ( 1 ) and ( 2 ),

$\Rightarrow$ $\angle TAP+\angle TBP=\angle PQA+\angle PQB$

But $△ATB,$

$\Rightarrow$ $\angle TAP+\angle TBP+\angle ATB={180}^o$

$\Rightarrow$ $\angle AQB={180}^o-\angle ATB$

$\Rightarrow$ $\angle AQB+\angle ATB={180}^o$

But they are the opposite angles of the quadrilateral.

$\therefore$ $AQBT$ are cyclic.

Hence, $A,Q,B$ and $T$ are concyclic