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:


Wikipedia – Thales’s theorem
畢達哥拉斯與泰利斯 – EpisteMath|數學知識
Wikipedia – Internal and external angles
Wikipedia – Thales of Miletus


◊ 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.

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