Consider three points $A, B$ and $C$ in the projective plane [the white points in the picture below] not all on one line. Next, choose a point $O$ [the blue point] and draw the lines that connect $O$ with $A, B$ and $C$ respectively. We will refer to these lines as the "blue lines". Then, we repeat this procedure with another point $O'$ in the plane [the red point in the picture]; the corresponding lines will be called the "red lines".
We then define the points of intersection of the blue line through A and the red line through B, the blue line through B and the red line through C, the blue line through C and the red line through A, and connect these intersection points [the purple points] via lines with $C, A$ and $B$ respectively. Prove that the latter three lines [the white lines in the picture] are concurrent.
I would like to obtain a proof of the statement above by constructing suitable (degenerate) cubics and/or conics, and applying classical theorems.