# Euclidean GeometryIntroduction

Mathematics has been studied for thousands of years – to predict the seasons, calculate taxes, or estimate the size of farming land.

Mathematicians in ancient Greece, around 500 BC, were amazed by mathematical patterns, and wanted to explore and explain them. For the first time, they began to study mathematics just “for fun”, without a specific application in mind.

One of these mathematicians was

Start by picking two points anywhere in the box on the left. Let’s draw a semicircle around these points.

Now pick a third point that lies somewhere on the circumference of the semicircle.

We can draw a triangle formed by the two corners of the semicircle, as well as the point you picked on the circumference.

Try moving the position of the three points and observe what happens to the angle at the top of the triangle. It seems like it is always

For Thales, this was a pretty spectacular result. Why should *semicircles* and *right-angled triangles*, two completely different shapes, be linked in this fundamental way? He was so awed by his discovery that, according to legend, he sacrificed an entire ox to thank the gods.

However, simply *observing* a relationship like this was not enough for Thales. He wanted to understand *why* it is true, and verify that it is *always* true – not just in the few examples he tried.

An argument that logically explains, beyond any doubt, why something must be true, is called a **proof***Thales’ theorem*.

But geometry is not just useful for proving theorems – it is everywhere around us, in nature, architecture, technology and design. We need geometry for everything from measuring distances to constructing skyscrapers or sending satellites into space. Here are a few more examples:

In this and the following courses, you will learn about many different tools and techniques in geometry, that were discovered by mathematicians over the course of many centuries. We will also see how these techniques can be used to solve important problems in the real world.