In this book we have approached plane geometry from the analytic point of view. The more traditional treatment of these topics (going back to Euclid) is based on geometric axioms or postulates and synthetic proofs (i.e., not using computation).

This approach is also instructive and complements our own. Essentially the same set of theorems can be derived, and the reader is encouraged to take the exercises from a book using the axiomatic approach and try to solve them by using our methods.

An excellent book using the axiomatic approach is that of Greenberg. Following Hilbert, whose work we mentioned in the Historical Introduction, he first introduces incidence axioms, which guarantee that two points are incident with a unique line, that every line is incident with at least two points, and that not all points are collinear. Then he introduces axioms for betweenness from which segment, ray, half-plane, angle, and its interior can be defined. These axioms are strong enough to allow one to prove Pasch's theorem and the crossbar theorem. The betweenness axioms are already too strong to be consistent with the geometry of the sphere or the projective plane, however. (A different set of betweenness axioms would be necessary.) Then an (undefined) relation of congruence on segments, angles, and triangles is introduced along with certain axioms that congruence is to satisfy.