0. Primitives
- Terms
- [[Point]]
- [[Line, Line-Segment, Ray]]
- [[Plane]]
- Relations
- Betweenness, a ternary relation linking points;
- example:
- Point $B$ is between $A$ and $C$
- Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes;
- examples:
- Point $P$ lies on line $\ell$
- Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an [[infix]] $\cong$
I. Incidence
1. For every two points $A$ and $B$ there exists a line $a$ that contains them both. We write $AB = a$ or $BA = a$. Instead of "contains", we may also employ other forms of expression; for example, we may say "$A$ lies upon $a
quot;, "$A$ is a point of $aquot;. "$a$ goes through $A$ and through $Bquot;, "$a$ joins $A$ to $Bquot;, etc. If $A$ lies upon $a$ and at the same time upon another line $b$, we make use also of the expression: "The lines $a$ and $b$ have the point $A$ in common", etc.
2. For every two points there exists no more than one line that contains them both; consequently, if $AB = a$ and $AC = a$, where $B\neq C$, then also $BC = a$.
3. There exist at least two points on a line. There exist at least three points that do not lie on the same line.
4. For every three points $A, B, C$ not situated on the same line there exists a plane $\alpha$ that contains all of them. For every plane there exists a point which lies on it. We write $ABC = \alpha$. We employ also the expressions: "$A, B, C$ lie in $/alphaquot;; "$A, B, C$ are points of $/alphaquot;, etc.
5. For every three points $A, B, C$ which do not lie in the same line, there exists no more than one plane that contains them all.
6. If two points $A, B$ of a line $a$ lie in a plane $\alpha$, then every point of $a$ lies in $\alpha$. In this case we say: "The line $a$ lies in the plane $\alphaquot;, etc.
7. If two planes $\alpha, \beta$ have a point $A$ in common, then they have at least a second point $B$ in common.
8. There exist at least four points not lying in a plane
### II. Order
1. If a point $B$ lies between points $A$ and $C,B$ is also between $C$ and $A$, and there exists a line containing the distinct points $A,B,C$.
2. If $A$ and $C$ are two points, then there exists at least one point $B$ on the line $AC$ such that $C$ lies between $A$ and $B$.
3. Of any three points situated on a line, there is no more than one which lies between the other two.
4. "Pasch's Axiom": Let $A$, $B$, $C$ be three points not lying in the same line and let $a$ be a line lying in the plane $ABC$ and not passing through any of the points $A$, $B$, $C$. Then, if the line $a$ passes through a point of the segment $AB$, it will also pass through either a point of the segment $BC$ or a point of the segment $AC$. (Less formally: a line entering a triangle in one direction will leave the triangle on another side)