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 2 of 5: Construct the second candidate circle

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r^2=TP\cdot TQ=TR\cdot TS

Detailed analysis

Let $\Gamma$ be the circle with center $T$ and radius $r$ , where $r^2=TP\cdot TQ=TR\cdot TS$ . For every point $V$ on $\Gamma$ , the power of $T$ with respect to the circle through $P,Q,V$ equals $TV^2$ , so $TV$ is tangent to that circle. The same argument applies to the circle through $R,S,V$ .