targas-edu.de
DE ☾ A+

Theorem of Thales

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

You learn: if a point \(C\) lies on the semicircle over \(\overline{AB}\), the angle at \(C\) is a right angle. You see the proof and construct a right-angled triangle.
📄 Open worksheet (PDF, German)

The video

In the video the point \(C\) moves along the semicircle. The angle at \(C\) always stays \(90^\circ\). Then come the proof and a construction with compasses.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:16 Thales' theorem
  3. 0:42 Why?
  4. 1:34 Constructing a triangle
  5. 2:29 Careful, mistake!
  6. 2:58 Try it yourself

The most important points

Thales' theorem (der Satz des Thales)

If the point \(C\) lies on the semicircle over \(\overline{AB}\) (the Thales circle, der Thaleskreis), the angle at \(C\) is a right angle: \(\gamma = 90^\circ\).

C moves along the semicircle. The angle at C stays 90^\circ.
\(C\) moves along the semicircle. The angle at \(C\) stays \(90^\circ\).

Proof

\(\overline{MA} = \overline{MB} = \overline{MC}\) (radii). The triangles \(AMC\) and \(MBC\) are isosceles, the base angles are \(\alpha\) and \(\beta\). Angle sum: \(\alpha + \beta + (\alpha + \beta) = 180^\circ\), so \(\alpha + \beta = 90^\circ\).

Construction: semicircle around M, arc around A with b = 4 cm.
Construction: semicircle around \(M\), arc around \(A\) with \(b = 4\) cm.
Mistake: C lies outside the circle. Then the angle is smaller than 90^\circ.
Mistake: \(C\) lies outside the circle. Then the angle is smaller than \(90^\circ\).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
Page 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

\(C\) lies on the circle with diameter \(\overline{AB}\). \(\alpha = 35^\circ\). How big is \(\beta\)?

Show solution

\(\gamma = 90^\circ\), so \(\beta = 180^\circ - 90^\circ - 35^\circ = 55^\circ\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Satz des Thales (Halbkreis, rechter Winkel, Thaleskreis, Beweis mit gleichschenkligen Dreiecken). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Use simple German sentences.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.