6.21Any two perspective triangles are related by a perspective collineation, namely an elation or a homology according as the center and axis are or are not incident.
Take two triangles PQR and P’Q’R’ perspective from O. By 6.13, there is just one projective collineation that transforms the quadrangle DEPQ into DEP’Q’. This collineation, transforming the line o=DE into itself, and PQ into P’Q’, leaves invariant the point F. By axiom 2.18, it leaves invariant every point on o. The join of any two distinct corresponding points meets o in an invariant point, and is therefore an invariant line.
The two invariant lines PP’ and QQ’ meet in an invariant point, namely O. The point R is transformed into R’. By the dual of axiom 2.18, every line through O is invariant.