Let $\triangle ABC$ be a triangle, and $\Gamma$ be a circle with center $O$ passing through $A$, which intersects $[AB]$ in $K$, $[BC]$ in $L$ and $M$ such that $L$ is between $B$ and $M$, $[AC]$ in $N$. Let $U$ be the center of the circle circumscribed of $\triangle KBL$ and $V$ be the center of the circle circumscribed of $\triangle NCM$.
How can we show that $(UL)$ and $(VM)$ intersect on $\Gamma$ ?
I tried, but I can't quite express the fact that the intersection of $(UL)$ and $(VM)$ is on $\Gamma$, except that the power of this point relative to $\Gamma$ is zero, or that the intersection of $(UL)$ and $\Gamma$, $V$ and $M$ are aligned. However, I can't find any properties involving this. I have also tried to deal with the problem analytically, but I don't think it is particularly relevant, because there is no obvious orthonormal marker that could be used to write the equations of circles, for example.