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Euclidean GeometryIntroduction

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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.

A Babylonian clay tablet, dated 1800 BC, that contains geometric calculations.

One of these mathematicians was Thales of Miletus, who made a surprising discovery when playing around with geometric shapes:

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.

Let’s draw the 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 °! This means that the triangle is right-angledequilateralacute.

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. In the following courses you will learn a number of geometric techniques, that will eventually allow us to prove 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:

Geometry allowed the ancient Egyptians to construct gigantic, perfectly regular pyramids.

Sailors use sextants to determine their location while at sea, using angles formed by the sun or stars.

Geometry is needed to create realistic video game or movie graphics.

Geometry can help design and test new airplane models, making them safer and more efficient.

Geometry was key when designing this skyscraper in Beijing – and to make sure it won’t fall over.

Geometry allows us to predict the position of stars, planets and satellites orbiting Earth.

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.