Coxeter- Theorem 2.24

2.24 There exist four coplanar points of which no three are collinear. PROOF. By our first three axioms, there exist two distinct lines having a common point and each containing at least two other points, say lines EA and EC containing also B and D, respectively, as in the figure provided. The four distinct points A,B,C,D have the desired property of noncollinearity. For instance, if the three points A,B,C were collinear, E (on AB) would be collinear with all of them, and EA would be the same line as EC, contradicting our assumption that these two lines are distinct.