Strange Problem

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Let the acute-angled triangle $ABC$ is inscribed in a circle $k$. Internally for $ABC$ is chosen a point $P$. Line $AP$ intersects $k$ at the point $A_1$. Line $BP$ intersects $k$ at the point $B_1$. Through $P$ is drawn a line $l$ intersecting $BC$ and $AC$ at the points $A_2$ and $B_2$. Prove that circumcircles of the triangles $A_1A_2C$, $B_1B_2C$ and the line $l$ intersects at a common point.