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 $\map b {A, B, C}$)


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


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

The structure $\EE = \tuple {P, L, I, b, \cong}$ 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 $p$:


 * $\tuple {p, l} \in I$

Then we say that $l$ contains $p$.

Collinearity
Let $S \subseteq P$.

Suppose that there exists a line $l$ containing all points in $S$::


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

Then $S$ is said to be collinear.

Segment
Let $A, B \in P : A \ne B$

Let $\overline{AB}$ be the set of $A$, $B$, and points between $A$ and $B$:


 * $\overline{AB} = \set {C \in P : \map b {A, C, B} \lor \paren {C = A} \lor {C = B}}$

Then the set $\overline{AB}$ is called the segment.

Congruence of Segments
Let $\overline {AB}, \overline {A'B'} \subseteq P$ be segments.

Let $\overline {AB}, \overline {A'B'}$ be congruent.

Then we denote this by $\overline{AB} \cong \overline{A'B'}$.

Ray
Let $A, B \in P : A \ne B$.

Let $S$ be the set of points $C$ such that $C$ is between $A$ and $B$ or $B$ is between $A$ and $C$:


 * $S = \set {C \in P : \map b {A, C, B} \lor \map b {A, B, C} }$

Let $\buildrel \to \over {A B}$ be the set $S$ extended with $A$ and $B$:


 * $\buildrel \to \over {A B} = \set {A, B} \cup S$

Then the set $\buildrel \to \over {A B}$ is called the ray.

Interior Point of Ray
Let $\buildrel \to \over {A B}$ be a ray.

Suppose $p \in P : \neg \map I {p, \buildrel \to \over {A B} } \land p \ne A$.

Then $p$ is called the interior point (of $\buildrel \to \over {A B}$).

Angle
Let $A, O, B \in P$.

Suppose $\set {A, O, B}$ are not collinear.

Let $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$ be rays.

Let $\angle AOB = \buildrel \to \over {O A} \cup \buildrel \to \over {O B}$

Then $\angle AOB$ is called the angle (between $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$).

Congruence
Let $\angle ABC$ and $\angle A'B'C'$ be angles.

Suppose $\angle ABC$ and $\angle A'B'C'$ are congruent.

Then we denote this by $\angle ABC \cong \angle A'B'C'$.

Points on the same side
Let $l \in L$.

Suppose $A, B \in P : \neg \map I {A, l} \land \neg \map I {B, l}$.

Let $\overline {AB}$ be a segment.

Suppose:


 * $\forall p \in \overline {AB} : \neg \map I {p, l}$

Then we say that $A$ and $B$ are on the same side of $l$.

Parallel lines
Let $l_1, l_2 \in L$.

Suppose no point lies on both $l_1$ and $l_2$:


 * $\neg \exists p \in P : \paren {l_1 \text { contains } p} \land \paren{l_2 \text { contains } p}$

Then $l_1$ and $l_2$ are said to be parallel.

Incidence postulates

 * a)

Let $A, B \in P$.

Then:


 * $\exists ! l \in L : \paren {l \text { contains } A} \land \paren {l \text { contains } B}$


 * b1)


 * $\forall l \in L : \exists A, B \in P : A \ne B : \map I {A, l} \land \map I {B, k}$


 * b2)

Let $A, B, C \in P$.

Let $S = \set {A, B, C}$

Then:


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

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 $l$.