Hilbert’s Axioms for Euclidean Plane Geometry

Undefined Terms

point, line, incidence, betweenness, congruence

Axioms

1. Axioms of Incidence

Postulate I.1.

For every point P and for every point Q not equal to P, there exists a unique line $$\ell$$ incident with the points P and Q.

Postulate I.2.

For every line $$\ell$$ there exist at least two distinct points P and Q incident with $$\ell$$.

Postulate I.3.

There exist at least three distinct points with the property that no line is incident with all three of them.

II.    Axioms of Betweenness

Postulate B.1.

If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B then also lies between C and A.

Postulate B.2.

For two distinct points B and D, there exist points A, C, and E on the line BD such that B lies between A and D, C lies between B and D, and D lies between B and E.

Postulate B.3.

If A, B and C are three distinct points lying on the same line, then one and only one of the points lies between the other two.

Postulate B.4. (Plane Separation Axiom)

For every line $$\ell$$ and for any three points A, B, and C that do not lie on $$\ell$$:

a)     If A and B are on the same side of $$\ell$$ and B and C are on the same side of $$\ell$$, then A and C are on the same side of $$\ell$$.

b)     If A and B are on opposite sides of $$\ell$$ and B and C are on opposite sides of $$\ell$$, then A and C are on the same side of $$\ell$$.

1. Axioms of Congruence

Postulate C.1.

If A and B are distinct points and A' is a point, then for each ray r emanating from A' there is a unique point B' on r such that $$B'\ne A'$$ and such that AB and A'B' are congruent.

Postulate C.2.

If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and A"B" are congruent to each other.

Postulate C.3.

On a line m, let AB and BC be two segments which, except for B, have no points in common. Furthermore, on the same or another line m', let A'B' and B'C' be two segments which, except for B', have no points in common. In that case if $$AB\cong A'B'$$ and $$BC\cong B'C'$$, then $$AC\cong A'C'$$.

Postulate C.4.

If $$\angle BAC$$ is an angle and if $$\overrightarrow{B'C'}$$ is a ray, then there is exactly one ray $$\overrightarrow{B'A'}$$ on each side of line $$\overleftrightarrow{B'C'}$$ such that $$\angle B'A'C' \cong \angle BAC$$.

Postulate C.5.

If $$\angle A \cong \angle B$$ and $$\angle A \cong \angle C$$, then $$\angle B \cong \angle C$$. Moreover, every angle is congruent to itself.

Postulate C.6. (SAS)

If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

1. Axioms of Continuity

Archimedes Axiom

If AB and CD are any segments, then there exists a number n such if segment CD is laid off n times on the ray $$\overrightarrow {AB}$$ emanating from A, then a point E is reached where $$n \cdot CD \cong AE$$ and B is between A and E.

Dedekind’s Axiom

Suppose that the set of all points on a line $$\ell$$ is the union $$\Sigma_1 \cup \Sigma_2$$ of two nonempty subsets such that no point of $$\Sigma_1$$ is between two points of $$\Sigma_2$$ and vice versa. Then there is a unique point, O, lying on $$\ell$$ such that O liest between $$P_1$$ and $$P_2$$ if and only if $$P_1 \in \Sigma_1$$ and $$P_2 \in \Sigma_2$$ and $$O \ne P_1,\,P_2$$.

The following two Principles follow from Dedekind's Axiom, yet are at times more useful.

Elementary Continuity Principle:

If one endpoint of a segment is inside a circle and the other is outside, then the segment intersects the circle.

Circular Continuity Principle:

If one circle has one point inside and one point outside another circle, then the two circles intersect in two points.

1. Axiom of Parallels

Let $$\ell$$ be any line and P a point not on it. Then there is at most one line that passes through P and does not intersect $$\ell$$.