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 derived 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)


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


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

the composite relations are:


 *  insideness  (binary)(point between segment endpoints)

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 $p$.

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 $\overline {AB}, \overline {A'B'} \subseteq P$ 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 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 $A$, $C$ be distinct points.

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


 * $A \ne C \implies \exists B : \map {\operatorname{between} } {A, C, B}$


 * $\exists B \in P : \map b {A, C, B}$

c)

Let $l \in L$ be a line.

Let $A, B, C \in P$ be distinct points on $l$:


 * $\map I {A, l} \land \map I {B, l} \land \map I {C, l}$

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 $BC$:


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

Congruence Postulates
a)

Let $l, l' \in L$ be lines.

Let $A, B \in P$ be distinct points on $l$:


 * $\map I {A, l} \land \map I {B, l}$

Let $A' \in P : \map I {A', l'}$

Then:


 * $\exists B' \in P : \map I {B', l'} : \overline {AB} \cong \overline {A'B'}$

b)

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

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

c)

Let $l, l' \in L$ be lines.

Let $\overline{AB}, \overline{BC} \subseteq l$ be segments such that:


 * $\overline{AB} \cap \overline{BC} = \set B$

Let $\overline{A'B'}, \overline{B'C'} \subseteq l$ or $\overline{A'B'}, \overline{B'C'} \subseteq l'$ be segments such that:


 * $\overline{A'B'} \cap \overline{B'C'} = \set {B'}$

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

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

d)

Let $l' \in L$ be a line.

Let $\angle rs$ be an angle

Let $\SS$ be a definite side of $l'$.

Let $\vec {r'}$ be a ray on $l'$ starting at a point $O'$.

Then:


 * $\exists ! \vec{s'} : \angle r's' \cong \angle rs$ and all the interior points of $\vec {s'}$ lie on $\SS$.

e)

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

 * 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 $l$.