The picture illustrates the "cross-line-theorem' in a projective plane for a situation, where 3 points (A1,A2,A3) on a line a are in perspectivity to 3 points (B1,B2,B3) on another line b from a center Z not on a nor b. If the quadrangles A1A2B2B3 and A2A3B4B5 are fano (see de:Fano-Axiom, then the black "crosses" (A1B2,A2B1) and (A2B3,A3B2) have points of intersection F12, F23 NOT on the line ZF (F=ab). The coloured triangles are in perspective from Z, so if Desargues' theorem holds, their correspondig sides intersect on a common line f, which is then a line through F, F12 and F23. Then the perspectivity (central projectivity) with center Z and axis f maps the red triple on a onto the red triple on b and is a projectivity of the whole plane.
Own work
KlioKlein