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 4 of 5: Recover the candidate circle from any tangency point

Previous step Next step

TV\cdot TV_1=TP\cdot TQ=TR\cdot TS=TV\cdot TV_2\Longrightarrow V_1=V_2=V

Detailed analysis

Conversely, let $V\in\mathcal A$ . If the two circles coincide, they are $C$ and $V\in C$ . Otherwise let $TV$ meet the two circles again at $V_1,V_2$ . Power of $T$ gives $TV\cdot TV_1=TP\cdot TQ$ and $TV\cdot TV_2=TR\cdot TS$ . The equality from the first step and $T\ne V$ imply $TV_1=TV_2$ , hence $V_1=V_2=V$ ; therefore $TV$ is tangent and $TV^2=r^2$ .