Question

Two circles intersect in points 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, B and T lie on a circle.

Answer 1

It is a common property that you should remember that whenever such condition occur then AQ and BQ will pass through the center of the circle and hence

$\angle QAT=\angle QBT={90}^{0}inquadrilateralInAQBT,\angle QAT+\angle QBT=90+90={180}^{0}SoAQBTiscyclic.$