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 objects are:


 * point


 * line

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)


 * insideness (binary)(point between segment endpoints)


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

Definition of Collinearity
Let $A_1, A_2, \ldots $ be distinct points.

Let $L$ be a line.

Suppose $A_1, A_2, \ldots $ lie on $L$.

Then $A_1, A_2, \ldots $ are said to be collinear.

Definition of 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 $B$:


 * $\map {\operatorname {segment} } {A, B} = \set {X : \paren {A \ne B} \land \paren { \paren{X = A} \lor \paren{X = B} \lor \map {\operatorname{between} } {A, X, B} } }$

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 $A$, $B$ be distinct points.

Let $p$ be a point such that $p = A$ or $p$ is between $A$ and $B$ or $B$ is between $A$ and $p$.

The the collection of all such $p$ is called the ray $\buildrel \to \over {A B}$:


 * $\buildrel \to \over {A B} = \set {p : \paren {p = A} \lor \paren { \map {\operatorname{between} } {A, p, B} } \lor \paren { \map {\operatorname{between} } {A, B, p} } }$

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 $\buildrel \to \over {A B}$):


 * $\ds \paren {p \text { lies on} \buildrel \to \over {A B} } \land \paren {p \ne A} \implies \paren {p \text { in} \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 $L$.

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

Suppose no point lies both on both $L$ and $L'$:


 * $\neg \exists P : \paren {P \text { lies on } L} \land \paren{P \text { lies on } L'}$

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

Incidence axiom 1
Let $A$ and $B$ be distinct points.

Then there exists only one line $L$ such that $A$ and $B$ lie on $L$:


 * $\forall A, B : A \ne B : \exists ! L : \paren {A \text { lies on } L} \land \paren {B \text { lies on } L}$

Incidence axiom 2
Let $L$ be a line.

Then there exist two distinct points $A$ and $B$ which lie on $L$:


 * $\forall L : \exists A, B : \paren {A \ne B} \land \paren {A \text { lies on } L} \land \paren {B \text { lies on } L}$

Incidence axiom 3
There are three distinct points which are not collinear:


 * $\exists A, B, C : \paren {A \ne B} \land \paren {A \ne C} \land \paren {B \ne C} \land \paren {A, B, C \text { are not collinear }}$

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 $A$:


 * $\map {\operatorname{between} } {A, B, C} \implies \paren {A \ne B} \land \paren {A \ne C} \land \paren {B \ne C} \land \paren {A, B, C \text { are collinear } } \land \map {\operatorname{between} } {C, B, A}$

Order axiom 2
Let $L$ be a line.

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

Then there is a point $B$ on $L$ such that $C$ lies between $A$ and $B$:


 * $\paren {A \ne C} \land \paren {A \text { lies on } L} \land \paren {C \text { lies on } L} \implies \exists B : \map {\operatorname{between} } {A, C, B} \land \paren {B \text { lies on } L}$

Order axiom 3
Let $L$ be a line.

Let $A$, $B$, $C$ be distinct points that lie on $L$.

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

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

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


 * $\paren { \paren {A, B, C \text { are not collinear } } \land \paren{A, B, C \text { do not lie on } L} \land \paren {D \text { lies on } L} \land \paren {D\text { is in } AB} } \implies \paren {\exists E : \paren {E \text { lies on } L } \land \paren{ \paren{E \text { is in } AB} \lor \paren{L \text { is in } AC} } }$

Congruence Axiom 1
Let $L$, $L'$ be lines.

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

Let $A'$ be a point on $L'$.

Then there is a point $B'$ such that $\overline {AB} \cong \overline {A'B'}$:


 * $\paren {A \ne B} \land \paren {A \text { lies on } L} \land \paren {B \text { lies on } L} \land \paren{A' \text { lies on } L'} \implies \exists B' : \paren{\paren {B' \text { lies on } L'} \land \paren { {\overline {AB} \cong \overline {A'B'} } } }$

Congruence Axiom 2
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 $B$:


 * $B : \paren{\overline{AB} \text { lies on } L} \land \paren{\overline{BC} \text { lies on } L} \land \paren {B \text { lies on }\overline{AB} } \land \paren {B \text { lies on }\overline{BC} }$

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


 * $B' : \paren{\paren{\paren{\overline{A'B'} \text { lies on } L} \land \paren{\overline{B'C'} \text { lies on } L}} \lor \paren{\paren{\overline{A'B'} \text { lies on } L'} \land \paren{\overline{B'C'} \text { lies on } L'}}} \land \paren {B' \text { lies on } \overline{A'B'} } \land \paren{B' \text { lies on } \overline{B'C'}}$

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.

Let $L'$ be a line.

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 $\angle ACB \cong \angle A'C'B'$.

Euclidean Parallel Postulate
Let $L$ be a line.

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

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