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Coxeter- Theorem 2.22

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.