# Coxeter- Theorem 2.22

- Author:
- AndrewLyon

2.22 -

*Any two coplanar lines have at least one common point.*PROOF. Let E be a point coplanar with the two lines but not on either of them. Let*AC*be one of the lines. Since the plane*ACE*is determined by the pencil of lines through*E*that meet*AC*, the other one of the two given lines may be taken to join two points on distinct lines of this pencil, say*B*on*EA*, and*D*on*EC*, as in the figure provided. According to Axiom 2.14, the two lines*AC*and*BD*have a common point.