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 Euclidean geometry.

In the following the primitive objects are:


 * points (denoted by $A, B, C \ldots$)


 * lines (denoted by $a, b, c \ldots$)


 * planes (denoted by $\alpha, \beta, \gamma \ldots$)

the composite objects are


 * segment


 * ray


 * angle

the primitive relations are:


 * distinct (binary)(defined for all objects)


 * betweenness (ternary)(point between two other points)


 *  lies on  (binary)(point lies on line, point lies on ray, segment lies on line)


 * congruence (binary)(defined for segments and angles).

the composite relations are:


 * collinear ($n$-ary)(defined for points


 * coplanar ($n$-ary)(defined for points)


 * insideness (binary)(point between segment endpoints)


 * intersection (binary)(line, ray, segment)

Collinearity
Let $A, B, \ldots $ be distinct points.

Let $a$ be a line.

Suppose $A, B, \ldots $ lie on $a$.

Then $A, B, \ldots $ are said to be collinear.

Coplanarity
Let $A, B, \ldots $ be distinct points.

Let $\alpha$ be a plane.

Suppose $A, B, \ldots $ lie on $a$.

Then $A, B, \ldots $ are said to be coplanar.

Segment
Let $A$, $B$ be distinct points.

Then by segment we mean all the points which are same as $A$ or $B$ or are between $A$ and $B$.

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

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

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

Ray
Let $O$, $A$ be distinct points.

Let $P$ be a point such that $P = O$ or $P$ is between $O$ and $B$ or $B$ is between $O$ and $P$.

The the collection of all such $P$ is called the ray $\buildrel \to \over {O A}$.

Intersection
Let $a$ and $b$ be distinct extended structures like segments, rays or lines with their types not necessarily matching.

Suppose there is a point $P$ that lies both on $a$ and $b$.

Then we say that $a$ and $b$ intersect.

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

Suppose $p$ is a point such that $p$ lies on $\buildrel \to \over {A B}$ and $p \ne A$.

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

Interior Point of Angle
Let $\angle AOB$ be an angle.

Let $A'$ be a point on the ray $\vec {OA}$.

Let $B'$ be a point on the ray $\vec {OB}$.

Let $P$ be a point between $A'$ and $B'$.

Then $P$ is called an interior point of $\angle AOB$.

Angle
Let $A$, $O$, $B$ be noncolinear distinct points.

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

Then the collection of $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$ is called the angle $\angle AOB$.

Congruence of Angles
Let $\angle ABC$ and $\angle A'B'C'$ be distinct 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 $l$ be a line.

Let $A$, $B$ be distinct points.

Suppose $A$ and $B$ do not lie on $l$.

Let $AB$ be a segment.

Suppose $AB$ does not intersect $l$.

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

Side of Line
Let $l$ be a line on the plane $\alpha$.

All points on $\alpha$ lying on the same side of $l$ are called the side (of $l$).

Parallel lines
Let $l, l'$ be distinct lines.

Suppose no point lies both on both $l$ and $l'$.

Then $l$ and $l'$ are said to be parallel.

Incidence axiom 1
Let $A, B$ be two distinct points.

Then there is only one line $a$ such that $A, B$ lie on $a$.

Incidence axiom 2
Let $a$ be a line.

Then there are two distinct points $A$ and $B$ which lie on $a$.

Incidence axiom 3
Let $A, B, C$ be three distinct noncollinear points.

Then there is only one plane $\alpha$ such that $A, B, C$ lie on $\alpha$.

Incidence axiom 4
Let $\alpha$ be a plane.

Then there are three noncollinear distinct points $A, B, C$ which lie on $\alpha$.

Incidence axiom 5
Let $A, B$ be two distinct points.

Let $a$ be the line on which $A, B$ lie.

Suppose $A, B$ lie on the plane $\alpha$.

Then every point of the line $a$ lies on $\alpha$.

Incidence axiom 6
Let $\alpha, \beta$ be distinct planes.

Suppose the point $A$ lies on $\alpha$ and $\beta$.

Then there is another point $B$ which lies on $\alpha$ and $\beta$.

Incidence axiom 7
Let $\alpha$ be a plane.

Then there are four distinct points which are not coplanar.

Order axiom 1
Let $A$, $B$, $C$ be points.

Suppose $B$ lies between $A$ and $C$

Then $A$, $B$, $C$ are distinct and collinear, and $B$ lies between $C$ and $A$.

Order axiom 2
Let $a$ be a line.

Let $A$, $C$ be distinct points on $a$.

Then there are points $B, D$ on $a$ such that $C$ lies between $A$ and $B$, and $C$ lies between $A$ and $D$.

Order axiom 3
Let $A$, $B$, $C$ be distinct collinear points.

Then no more than one of them lies between the other two.

Order axiom 4 (Axiom of Pasch)
Let $A$, $B$, $C$ be non-collinear points.

Let $a$ a line such that $A$, $B$, $C$ do not lie on $a$.

Suppose there is a point $D$ that lies on $a$ and is in the segment $AB$.

Then there is a point $E$ which lies on $a$ and is in $AC$ or $BC$.

Congruence Axiom 1
Let $l, l'$ be same or distinct lines.

Let $A, B$ be distinct points on $l$.

Suppose $A'$ is a point on $l'$.

Then there are two and only two points $B'$ and $C'$ on $l'$ such that all the following conditions hold:


 * $A'$ is between $B'$ and $C'$;


 * $\overline {AB} \cong \overline {A'B'}$;


 * $\overline {AB} \cong \overline {C'A'}$.

Congruence Axiom 2
Let $\overline {AB}, \overline {A'B'}, \overline { {A'}'{B'}'}$ be segments.

Suppose $\overline {A'B'} \cong \overline {AB}$ and $\overline { {A'}'{B'}'} \cong \overline {AB}$.

Then $\overline {A'B'} \cong \overline { {A'}'{B'}'}$

Congruence Axiom 3
Let $l, l'$ be distinct lines.

Let $\overline{AB}$, $\overline{BC}$ be segments on $l$ with a commont point $B$.

Let $\overline{A'B'}$, $\overline{B'C'}$ be segments both either on $l$ or $l'$ with a common point $B'$:

Suppose $\overline {AB} \cong \overline{A'B'}$ and $\overline{BC} \cong \overline{B'C'}$.

Then $\overline{AC} \cong \overline{A'C'}$.

Congruence Axiom 4
Let $\angle AOB$ be an angle on the plane $\alpha$.

Let $l'$ be a line on the plane $\alpha'$.

Let $\vec{O'A'}$ be a ray.

Suppose $\vec{O'A'}$ lies on $l'$.

Then there is only one ray $\vec{O'B'}$ such that $\angle AOB \cong \angle A'O'B'$ and all interior points of $\angle A'O'B'$ lie on the given side of $l'$.

Congruence Axiom 5
Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles.

Suppose:


 * $\overline{AB} \cong \overline{A'B'}$


 * $\overline{AC} \cong \overline {A'C'}$


 * $\angle BAC \cong \angle B'A'C'$

Then $\angle ABC \cong \angle A'B'C'$ and $\angle ACB \cong \angle A'C'B'$.

Congruence Axiom 6
Let $\angle AOB, \angle A'O'B', \angle {A'}'{O'}'{B'}'$ be angles.

Suppose $\angle AOB \cong \angle A'O'B'$ and $\angle AOB \cong \angle {A'}'{O'}'{B'}'$.

Then $\angle A'O'B' \cong \angle {A'}'{O'}'{B'}'$.

Euclidean Parallel Postulate
Let $l$ be a line on the plane $\alpha$.

Let $A$ be a point that does not lie on $l$.

Then there exists a unique line $l'$ on $\alpha$ such that $A$ lies on $l'$ and $l'$ is parallel to $l$.

Axiom of Continuity (Archimedes Axiom)
Let $\vec{AB}$ be a ray.

Let $A_1$ be a point between $A$ and $B$.

Let $A_2$ be a point such that $A_1$ lies between $A$ and $A_2$.

For $i\in\N_{>2}$ let $A_i$ be a new point such that $A_{i - 1}$ lies between $A_{i - 2}$ and $A_i$.

Suppose $AA_1 \cong A_1 A_2 \cong \ldots \cong A_i A_{i+1} \cong \ldots$.

Then there is $n \in \N_{>0}$ such that $B$ lies between $A_{n - 1}$ and $A_n$.

Axiom of Completeness
There are no other primitive objects that could be added to points, lines and planes while satisfying all of the axioms above.