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 5 of 5: State the complete locus

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\mathcal A=C\cup\Gamma,\qquad \Gamma=\{V:TV=\sqrt{TP\cdot TQ}\}

Detailed analysis

The forward and converse inclusions prove that the requested set is exactly $\mathcal A=C\cup\Gamma$ , where $\Gamma$ has center $T$ and radius $\sqrt{TP\cdot TQ}$ .