Axiom:Hilbert's Axioms

Definition
Hilbert's Axioms are a series of axioms whose purpose is to provide a rigorous basis for the definition of planar Euclidean geometry.

In the following the primitive notions are:


 * point


 * line

and the primitive relations are:


 * betweenness (the statement $B$ is between $A$ $C$ is denoted by $A*B*C$)


 * incidence (containment, lies on) denoted by $I \subseteq P \times L$; that is, if a point $p \in P$ is incident with a line $l \in L$ then $\tuple {p, l} \in I$


 * congruence (the statment $X$ is congruent to $Y$ is denoted by $X \cong Y$).

The structure $\EE = \tuple {P, L, I, b, \equiv}$ is called a Euclidean space if it satisfies the following axioms:

Containment
Let $l \in L$ and $p \in P$.

Suppose $l$ is incident with $p$:


 * $\map I {l, p}$

Then we say that $p$ lies on $l$.

Collinearity
Let $S \subseteq P$.

Suppose:


 * $\forall p \in S : \exists l \in L : \map I {p, l}$

Then $S$ is said to be collinear.

Segment
Given two distinct points $A$ and $B$, the segment $\overline{AB}$ is the set consisting of $A$, $B$, and all points $C$ such that $C$ is between $A$ and $B$

Congruence
The congruence of $\overline {AB}$ to $\overline {A'B'}$ is denoted by $\overline{AB} \cong \overline{A'B'}$.

Ray
Given two distinct points $A$, $B$, the ray $\buildrel \to \over {A B}$ is the set consisting of $A$, $B$, and all points $C$ such that either $C$ is between $A$ and $B$ or $B$ is between $A$ and $C$.

Interior point of Ray
An interior point of the ray $\buildrel \to \over {A B}$ is a point that lies on $\buildrel \to \over {A B}$ and is not equal to $A$.

Angle
Given three noncollinear points $A$, $O$, $B$, the angle $\angle AOB$ is the union of the rays $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$.

Congruence
The notation $\angle ABC \cong \angle A'B'C'$ means that $\angle ABC$ is congruent to $\angle A'B'C'$.

Points on the same side
Given a line $l$ and two points $A$, $B$ that do not lie on $l$, we say that $A$ and $B$ are on the same side of $l$ if no point of $\overline {AB}$ lies on $l$.

Parallel lines
Two lines are said to be parallel if there is no point that lies on both of them.

Incidence postulates

 * a) For any two distinct points $A$, $B$, there exists a unique line that contains both of them.


 * b) There exist at least two points on each line, and there exist at least three noncollinear points.

Order Postulates

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


 * b) Given two distinct points $A$ and $C$, there always exists at least one point $B$ such that $C$ lies between $A$ and $B$.


 * c) Given three distinct points on a line, no more than one of them lies between the other two.


 * d) Let $A$, $B$, $C$ be three noncollinear points, and let $l$ be a line that does not contain any of them. If $l$ contains a point of $\overline {AB}$, then it also contains a point of $\overline {AC}$ or $\overline {BC}$.

Congruence Postulates

 * a) If $A$, $B$ are two points on a line $l$, and $A'$ is a point on a line $l'$, then it is always possible to find a point $B'$ on a given ray of $l'$ starting at $A'$ such that $\overline {AB} \cong \overline {A'B'}$.


 * b) If segments $\overline {A'B'}$ and $\overline { {A'}'{B'}'}$ are congruent to the same segment $\overline {AB}$, then $\overline {A'B'}$ and $\overline { {A'}'{B'}'}$ are congruent to each other.


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


 * d) Let $\angle rs$ be an angle and $l'$ a line, and let a definite side of $l'$ be given. Let $\vec {r'}$ be a ray on $l'$ starting at a point $O'$. Then there exists one and only one ray $\vec{s'}$ such that $\angle r's' \cong \angle rs$ and at the same time all the interior points of $\vec {s'}$ lie on the given side of $l'$.


 * e) If for two triangles $\triangle ABC$ and $\triangle A'B'C'$ the congruences $\overline{AB} \cong \overline{A'B'}$, $\overline{AC} \cong \overline {A'C'}$, and $\angle BAC \cong \angle B'A'C'$ hold, then $\angle ABC \cong \angle A'B'C'$ and $\angle ACB \cong \angle A'C'B'$ as well.

Euclidean Parallel Postulate

 * Given a line $l$ and a point $A$ that does not lie on $l$, there exists a unique line that contains $A$ and is parallel to $l$.