In euclidean, similarity and affine geometry, we have the concept of betweenness. Given three collinear points, one lies between the other two. This is preserved by the appropriate transformations. The concept of line segment is based on betweenness.
In projective geometry, there is no corresponding concept.


It follows that we cannot talk of the segment of a pline defined by two ppoints on the pline. As a consequence, the definition of a triangle in projective geometry is rather different from that in the other geometries. If A,B,C are noncollinear ppoints, then the sides of ΔABC are the (complete) plines AB, BC and CA. Given three distinct plines, L,M,N, these define a triangle with vertices the intersections of the plines in pairs.


Another concept which can be defined in terms of betweenness is that of the interior of a triangle. The sides of a projective triangle are represented by three planes through O. These divide space into eight regions. There is no consistent way to choose any one of these as the interior.
example
In Π, the vertices embed as X(1,0,0), Y(0,1,0), Z(0,0,1).
In Φ, the vertices embed as X, Y and Z'(0,0,1).
On the other hand, we can define the interior of a projective conic 
