Points, Lines & Planes

Geometry starts with three ideas it does not define: point, line and plane.

A point marks a position and has no size. A line goes on forever in both directions and has no thickness. A plane is a flat surface stretching forever in every direction.

That sounds like dodging the question, and it is — on purpose. Every definition has to be written using simpler words, and eventually you run out of simpler words. So geometry picks three starting ideas, describes them, and defines everything else from there.

Notation matters here. A point is a capital letter, A. A line through A and B is written with a double arrow over AB. A segment is the piece between two points, with a plain bar over AB. A ray starts at one point and goes forever the other way.

Point, line and plane are the primitive (undefined) terms of Euclidean geometry. Defining them would require prior terms, so an axiomatic system instead names its primitives, states axioms describing how they behave, and derives everything else.

The core incidence axioms: two distinct points determine exactly one line; three non-collinear points determine exactly one plane; if two points lie in a plane, the entire line through them lies in that plane; two distinct planes meeting at all meet in a line.

Vocabulary that later theorems depend on: points are collinear if one line contains them all, and coplanar if one plane does. Two lines in space that are neither parallel nor intersecting are skew, which is only possible in three dimensions.

The distinction between a figure and its measure runs through the whole course. AB with a bar denotes the segment, an object; AB without one denotes its length, a number. Segments are congruent (≅), lengths are equal (=), and using the wrong symbol in a proof is a genuine error rather than a typo.