Segment

Dependencies: Point–line axiom, Ruler axiom. A derived definition: use “between” on a line to cut a finite piece from an infinite line.

Definition

Let $A$ and $B$ be distinct points on a line. The Point–line axiom axiom gives the unique line through them, and the Ruler axiom axiom orders its points like real numbers, making “between” precise.

All points of the line between $A$ and $B$, including $A$ and $B$, form segment $AB$. $A$ and $B$ are its endpoints; the other points are interior points.

A segment is a finite piece cut from a line. Its length is the distance $|AB|$ between its endpoints. The sides of triangles and polygons are segments.

Remarks

• Both endpoints are included. A segment is closed at both ends.
• Segment, ray, and line. A segment is finite with two endpoints; a ray has one endpoint and extends infinitely one way; a line extends infinitely in both directions.
• No direction. Segments $AB$ and $BA$ are the same point set. A directed segment or vector requires additional structure.
• Length and midpoint are derived from it. Once a segment is fixed, its length $|AB|$ and midpoint are determined. Medians, midsegments, and polygon perimeters are built from these ideas.