an undefined term

Try to define a word using only simpler words, then define those words too, and those — eventually you either go in a circle or run out of room. Every dictionary secretly leans on this: somewhere the explanations must stop. In geometry we make that stopping point honest and explicit by choosing a few words we agree never to define. These are the undefined terms, or primitive notions.

In Euclidean geometry the standard undefined terms are point, line, and plane (some treatments also leave 'lies on' or 'between' undefined). We do not say what a point is; instead the axioms say how points, lines, and planes relate — for instance, 'through any two points there is exactly one line'. From these primitives and axioms, every other geometric word is defined: a segment is part of a line between two points, an angle is two rays sharing an endpoint, and so on. The undefined terms are the bedrock everything else is built on.

The honest reason to leave terms undefined is not laziness but logic: a finite system that proves and defines everything is impossible, because each definition uses earlier words and each proof uses earlier facts, so something must come first without either. A powerful consequence is that the undefined terms can mean anything that obeys the axioms. Any collection of objects and relations that satisfies the geometry's axioms is a model of it — so in one model a 'point' is a dot, in another a pair of numbers, in another a great circle's pole. Geometry studies the pattern, not the props.