Euclidean GeometryEuclid’s Axioms

A point is a specific location in space. Points describe a position, but have no size or shape themselves. They are labelled using capital letters.

In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move. Continue

A line is a set of infinitely many points that extend forever in both directions. Lines are always straight and, just like points, they don’t take up any space – they have no width.

Lines are labeled using lower-case letters like a or b. We can also refer to them using two points that lie on the line, for example PQ↔ or QP↔. The order of the points does not matter. Continue

A line segment is the part of a line between two points, without extending to infinity. We can label them just like lines, but without arrows on the bar above: AB‾ or BA‾. Like, before the order of the points does not matter. Continue

A ray is something in between a line and a line segment: it only extends to infinity on one side. You can think of it like sunrays: they start at a point (the sun) and then keep going forever.

When labelling rays, the arrow shows the direction where it extends to infinity, for example AB→. This time, the order of the points does matter. Continue

A circle is the collection of points that all have the same distance from a point in the center. This distance is called the radius. Continue

These two shapes basically look identical. They have the same size and shape, and we could turn and slide one of them to exactly match up with the other. In geometry, we say that the two shapes are congruent.

The symbol for congruence is ≅, so we would say that A≅B.

Two straight lines that never intersect are called parallel. They point into the same direction, and the distance between them is always the sameincreasingdecreasing.

A good example of parallel lines in real life are railroad tracks. But note that more than two lines can be parallel to each other!

In diagrams, we denote parallel lines by adding one or more small arrows. In this example, a∥b∥c and d∥e. The ∥ symbol simply means “is parallel to”.

The opposite of parallel is two lines meeting at a 90° angle (right angle). These lines are called perpendicular.

In this example, we would write a ⊥ b. The ⊥ symbol simply means “is perpendicular to”.

Greek mathematicians realised that to write formal proofs, you need some sort of starting point: simple, intuitive statements, that everyone agrees are true. These are called axioms (or postulates).

A key part of mathematics is combining different axioms to prove more complex results, using the rules of logic.

The Greek mathematician Euclid of Alexandria, who is often called the father of geometry, published the five axioms of geometry:

Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. According to none less than Isaac Newton, “it’s the glory of geometry that from so few principles it can accomplish so much”.

Euclid published the five axioms in a book “Elements”. It is the first example in history of a systematic approach to mathematics, and was used as mathematics textbook for thousands of years.