Points – The Starting Block of All Geometry

A point is the most fundamental object in all of geometry. Everything else — lines, angles, shapes — is built from points. Understanding what a point is, and how to work with points, is the first step in any geometry course.

In his Elements (c. 300 BCE), Euclid opened Book I with the definition “a point is that which has no part” — an idea so abstract that a point cannot truly be drawn, only represented by a dot far larger than the concept it stands for. Points remained a purely visual idea for almost 2,000 years until 1637, when the French mathematician and philosopher René Descartes published La Géométrie, introducing the (x, y) coordinate system that lets every point be written as a pair of numbers. This single idea fused algebra and geometry together and made possible everything from graphing calculators to computer-aided design. Today, the same principle locates you on a map: a GPS receiver calculates your position as a point described by two coordinates — latitude and longitude.

What Is a Point?

A point is an exact location in space. It has no size, no width, no length, and no thickness. It is simply a position. In diagrams, a point is shown as a small dot and labelled with a capital letter such as A, B, or P.

Notation

Symbol	Meaning	Example
A	A single point labelled A	Point A is at position (3, 2)
(x, y)	Coordinates on a grid	(5, -1) means 5 right, 1 down
(x, y, z)	Coordinates in 3D space	(2, 4, 1) in three dimensions

Points on a Coordinate Grid

When we place geometry on a number grid (the coordinate plane), every point can be described by a pair of numbers (x, y). The x-value tells you how far to go right or left; the y-value tells you how far to go up or down.

Plot point P(4, 3).

Start at the origin (0, 0). Move 4 units to the right, then 3 units up. Mark the location and label it P.

Plot point Q(-2, 5).

Start at the origin. Move 2 units to the left, then 5 units up. Mark and label Q.

Collinear Points

Three or more points are collinear if they all lie on the same straight line. If they do not, they are non-collinear.

Are A(1, 2), B(3, 4), and C(5, 6) collinear?

Check if the gradient between each pair is the same. A to B: (4-2)/(3-1) = 1. B to C: (6-4)/(5-3) = 1. Same gradient, so yes, they are collinear.

Distance Between Two Points

The distance between point A(x₁, y₁) and point B(x₂, y₂) is found using the distance formula, which comes directly from Pythagoras' theorem:

Distance = square root of [(x₂ − x₁)² + (y₂ − y₁)²]

Find the distance between A(1, 1) and B(4, 5).

Distance = √[(4−1)² + (5−1)²] = √[9 + 16] = √25 = 5 units.

Key Takeaways

A point is an exact position in space with no size or dimension.
Points are labelled with capital letters and located by coordinates (x, y).
Collinear points all lie on the same straight line.
The distance between two points uses the distance formula based on Pythagoras.

Practice: Distance & Collinearity

Find the Distance Between Two Points