Foundations of Geometry

Points, Lines & Planes

The undefined terms of geometry and how they relate.

Foundational20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Ideas So Basic They Can't Be Defined

Try to define the word 'point' without using the word 'point' or any close synonym like 'location' or 'spot.' Before reading on, actually attempt it. Notice how every definition you try seems to lean on some other undefined idea — 'a place with no size' still needs 'place' to already make sense.

This isn't a failure of imagination — it's a genuine limit of how definitions work. Every definition explains one idea using other, simpler words, but that chain can't go on forever without circling back on itself. Geometry solves this by starting from a small set of undefined terms, accepted as intuitively understood, and building every other definition rigorously on top of them.

Definition — Point, Line, and Plane

A point is an exact location with no size, usually named with a capital letter. A line is a straight path of points extending infinitely in two directions, named by two points on it or a lowercase letter. A plane is a flat surface extending infinitely in every direction, named by three points on it that don't all lie on one line. All three are undefined terms — described, not formally defined.

From these undefined terms, real definitions and facts follow. Points that lie on the same line are collinear; points that lie on the same plane are coplanar. Two accepted starting facts (called postulates, statements taken as true without proof) govern how these objects relate: exactly one line passes through any two distinct points, and exactly one plane passes through any three noncollinear points.

Worked Example — Identifying Collinear and Coplanar Points

Points A, B, and C lie on the same straight line; point D lies off that line but on the same flat surface as all three. A, B, and C are collinear (and therefore automatically coplanar too, since three points determine a plane). D is coplanar with A, B, and C, but not collinear with them.

Worked Example — Applying the Two-Points-Determine-a-Line Postulate

Two distinct planes intersect. What must their intersection look like, and why? Since a plane is flat and extends infinitely, two distinct planes can only meet along a straight line (not a curve, and not just a single point) — that shared line contains every point common to both planes.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

When a problem says three or more points are 'noncollinear,' it's really telling you something useful: those points determine a genuine plane, not just an infinite family of planes all containing the same line.

Common Mistakes

Assuming any three points determine a plane, without checking whether they're collinear first.
Three collinear points lie on infinitely many different planes, not just one — the 'three points determine a plane' postulate specifically requires the points to be noncollinear.
Treating 'coplanar' and 'collinear' as interchangeable, or assuming coplanar points must also be collinear.
Collinear means on the same line (a stronger condition); coplanar means on the same flat surface, which includes points that aren't in a straight line at all — collinear points are always coplanar, but not the reverse.

Key Takeaways

Point, line, and plane are undefined terms — geometry's necessary starting point, since not every word can be defined using simpler words forever.
Two points determine exactly one line; three noncollinear points determine exactly one plane.
Two distinct planes that intersect always meet along a single straight line.

Summary

Starting from undefined terms and a few basic postulates, geometry builds every other idea on solid logical ground. The next lesson uses these building blocks to study how angles relate to each other.

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