# What is Thales’ theorem? The little relationship between circle and right triangle.

Maybe some people would say that “who is Thales? I have only heard about Taylor Swift.” But you must have heard the theorem which I will talk about later.

## Who is Thales?

Thales was born in 624 B.C. He is a Greek philosopher, mathematician and was also one of the Seven Sages of Greece.He was known as “the father of philosophy” or “the father of mathematics” nowadays.

## What is Thales’ theorem talking about?

In short, The triangle formed by the diameter of a circle and a point on the circumference must be a right triangle.
Precisely, the angle opposite the diameter will be a right angle.

Now we speak in mathematical language:
Let A, B, C be three distinct points on a circle and the line AB be the diameter. Then $$\angle ACB$$ must be equal to $$90 ^{\circ}$$.

Try moving the orange dot along the circle, you can see the angle is really 90 degree:

## How to prove it?

The way to prove is very simple, just add one more line and you can see it.

Draw a line from C and to O. The diameter is fixed, so we know both two triangle $$\triangle AOC$$ and $$\triangle COB$$ is isosceles triangle.
Do you remember the characteristics of the isosceles triangle? $$\angle CAO = \angle OCA$$, and $$\angle OBC = \angle BCO$$.

To be easily reading, we call $$\angle CAO \alpha$$ and $$\angle OBC \beta$$.

We know the internal angle of triangle is $$180 ^{\circ}$$, so we try to write down the internal angle of $$\triangle ABC$$ by $$\alpha$$ and $$\beta: \alpha + \alpha + \beta + \beta = 2(\alpha+\beta)=180^{\circ}$$

Then we get $$\alpha+\beta=90^{\circ}$$, which is the angle we are talking about ( $$\angle ACB$$ ).

If you think it’s hard to imagine it, you can try to move the orange dot again and observe the change of every angles:

##### 英文調味

◊ circle – 圓
◊ center of circle – 圓心
◊ triangle – 三角形
◊ isosceles triangle – 等腰三角形
◊ angle – 角
◊ internal angle – 內角
◊ external angle – 外角
◊ right triangle – 直角三角形

Pictures credits:
All the graphs on Geogebra were made by Alex.
The Thales’ portrait is from Wikipedia.