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 3 of 5: Prove every point of the candidate circle is a tangency point

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V\in\Gamma\Longrightarrow TV\text{ is tangent to both circles through }(P,Q,V)\text{ and }(R,S,V)

Detailed analysis

The two circles through $(P,Q,V)$ and $(R,S,V)$ are both tangent to the same line $TV$ at $V$ . Hence they are tangent to each other at $V$ . Thus every point of $\Gamma$ belongs to $\mathcal A$ . Every point of $C$ also belongs to $\mathcal A$ , because in that case the two circles may both be $C$ .