# Coxeter- Theorem 6.21

- Author:
- LexiPasi

**6.21**

*Any 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.