Školjka

%V0 Let $ABC$ be a triangle with circumcircle $\omega$ and $l$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $l$. The side-lines $BC,CA,AB$ intersect $l$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.