I've been reading up on projective geometry as part of my preparation for Math Olympiads, and I was wondering whether all results of projective geometry can be used in Euclidean geometry, after, of course, accounting for the existence of parallel lines that don't meet in the Euclidean plane by extending the Euclidean plane to include a line at infinity.
More specifically, does this axiom of projective geometry hold in Euclidean geometry -
The three diagonal points of a complete quadrangle are never collinear.
if we define a quadrangle and its diagonals as defined in projective geometry, after extending the Euclidean plane. What about the other axioms?