Problem 3

Let

PQRS

be a cyclic quadrilateral such that the lines

PQ

and

RS

are not parallel. Let

\mathcal A

be the set of points of tangency of a circle through

P,Q

and a circle through

R,S

. Determine

\mathcal A

Step 1 of 5: Introduce the external intersection and its power

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T=PQ\cap RS,\qquad TP\cdot TQ=TR\cdot TS

Detailed analysis

Let $T=PQ\cap RS$ . Since $P,Q,R,S$ lie on the same circle $C$ , the intersecting-secant theorem gives $TP\cdot TQ=TR\cdot TS$ . The hypothesis that the lines are not parallel makes $T$ finite and outside $C$ .