A.2 Hilbert’s Axioms¶
The system begins with undefined terms point, line and plane and the fundamental incidence relation. We need two more relations: the betweenness relation denoted by \(*\), and the congruence relation denoted by \(\cong\). All three relations are undefined and are given meaning within the corresponding set axioms. A quick note about incidence: although the relation \(P\in l\) should, strictly speaking, be read: “\(P\) and \(l\) are incident,” we shall use “\(l\) contains \(P\),” “\(P\) lies on \(l\)” or any obviously logical equivalent.
Incidence Axioms¶
For every pair of distinct points \(A\) and \(B\) there is a unique line \(l\) containing \(A\) and \(B\).
Every line contains at least two points.
There are at least three points that do not lie on the same line.
Betweenness Axioms¶
The next group of axioms deals with the relation “\(B\) lies between \(A\) and \(C\).” We will use the notation \(A*B*C\) for “\(B\) lies between \(A\) and \(C\).”
If \(A*B*C\), then \(A\), \(B\) and \(C\) are distinct points on a line, and \(C*B*A\) also holds.
Given two distinct points \(A\) and \(B\), there exists a point \(C\) such that \(A*B*C\).
If \(A\), \(B\) and \(C\) are distinct points on a line, then one and only one of the relations \(A*B*C\), \(B*C*A\) and \(C*A*B\) is satisfied.
Let \(A\), \(B\) and \(C\) be points not on the same line and let \(l\) be a line which contains none of them. If \(D\in l\) and \(A*D*B\), there exists an \(E\) on \(l\) such that \(B*E*C\), or an \(F\) on \(l\) such that \(A*F*C\), but not both.
If we think of \(A\), \(B\) and \(C\) as the vertices of a triangle, another formulation of B4 is this: If a line \(l\) goes through a side of a triangle but none of its vertices, then it also goes through exactly one of the other sides. This formulation is also called Pasch’s Axiom. Note that this is not true in \(\mathbb{R}^n\) where \(n\geq3\). Hence, I.3 and B.4 together define the geometry as “2-dimensional.”
Congruence Axioms¶
We have two sets of three congruence axioms, the first for congruent line segments, the second for angle congrunce.
Given a segment \(AB\) and a ray \(r\) from \(C\), there is a uniquely determined point \(D\) on \(r\) such that \(CD\cong AB\).
\(\cong\) is an equivalence relation on the set of segments.
If \(A*B*C\) and \(A'*B'*C'\) and both \(AB\cong A'B'\) and \(BC\cong B'C'\), then also \(AC\cong A'C'\).
Given a ray \(\overrightarrow{AB}\) and an angle \(\angle B'A'C'\), there are angles \(\angle BAE\) and \(\angle BAF\) on opposite sides of \(\overrightarrow{AB}\) such that \(\angle BAE \cong \angle BAF \cong \angle B'A'C'\).
\(\cong\) is an equivalence relation on the set of angles.
Given triangles \(\triangle ABC\) and \(\triangle A'B'C'\). If \(AB \cong A'B'\), \(AC\cong A'C'\) and \(\angle BAC\cong \angle B'A'C'\), then the two triangles are congruent: \(BC\cong B'C'\), \(\angle ABC \cong \angle A'B'C'\) and \(\angle BCA \cong \angle B'C'A'\).