Team Selection Test for EGMO 2023

6. First $XY\parallel l_0$ implies that $XYBC$ is concyclic. Now let $S$ be the second intersection point of the circle ($XYZ$) with the circumcircle; then the lines $ZS,XY$ and $BC$ are concurrent since they are the pairwise radical axes of the circles ($XYBC$) and ($XYZS$) and the circumcircle (let $P$ denote the point of concurrence). Now the quadruple ($AP,AZ;AB,AC$) is harmonic, hence ($ZP,ZA;ZB,ZC$) is harmonic, thus ($S,A;B,C$) is harmonic, so $S$ is a fixed point.