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Perpendicular bisector and circumcircle

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

You construct the perpendicular bisector of a line segment with compasses. With it you also find the circumcircle of a triangle.
📄 Open worksheet (PDF, German)

The video

In the video the compasses draw two arcs. The crossing points lie on the perpendicular bisector. At the end a circle through all three corners appears.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:14 Same distance
  3. 0:32 Construction with compasses
  4. 1:36 The circumcircle
  5. 2:28 Careful, mistake!
  6. 2:47 Try it yourself

The most important points

Perpendicular bisector

The perpendicular bisector (die Mittelsenkrechte) \(m\) goes through the middle \(M\) of the segment \(\overline{AB}\) and stands perpendicular to it. Every point on \(m\) has the same distance from \(A\) and from \(B\).

The perpendicular bisector of \overline{AB}, constructed with compasses (der Zirkel).
The perpendicular bisector of \(\overline{AB}\), constructed with compasses (der Zirkel).

How to construct it

1. Set the compasses to more than half the segment. 2. Arc around \(A\). 3. Arc around \(B\) with the same radius. 4. Join the crossing points \(P\) and \(Q\).

Circumcircle

The perpendicular bisectors of the three sides of a triangle cross at one point \(U\). \(\overline{UA} = \overline{UB} = \overline{UC}\). The circle around \(U\) through \(A\), \(B\) and \(C\) is the circumcircle (der Umkreis).

The circumcircle goes through A, B and C.
The circumcircle goes through \(A\), \(B\) and \(C\).
Mistake: the compasses are set to 2 cm, less than half the segment. The arcs do not cross.
Mistake: the compasses are set to \(2\) cm, less than half the segment. The arcs do not cross.

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
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

A triangle has a right angle at \(C\). Where is the centre of the circumcircle?

Show solution

In the middle of the hypotenuse \(\overline{AB}\) (more in the video on the theorem of Thales).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Mittelsenkrechte und Umkreis (Konstruktion mit Zirkel, Abstand, Mittelpunkt des Umkreises). 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.