I have recently found this interesting question:
Q: Let $ABCD$ be a convex quadrilateral with sides having different lengths.
Two distinct points $P,Q$ are said a happy couple if $\widehat{APB}=\widehat{AQB}=\widehat{CPD}=\widehat{CQD}$.
Prove that all the lines defined by a happy couple share a common point.
I have managed to find a proof through powers and cross-ratios, but I am not entirely satisfied by it and I feel that a more elementary solution eluded me, which is the reason for posing this question. Some considerations:
- A happy couple defines / is defined by a couple of circles with chords $AB,CD$ which are mapped into each other by a spiral similarity;
- The line defined by a happy couple is precisely the radical axis of the previous circles and the radical axis of $\Gamma_{AB},\Gamma_{CD}$ is the locus of points $R$ such that $\text{Pow}(R,\Gamma_{AB})=\text{Pow}(R,\Gamma_{CD})$.