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← Level 5 – Equation kid

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Thales' theorem

Every triangle on the diameter of a circumference has a right angle. Why it holds and how to draw with it.

A right angle for free

Draw a circumference and its diameter AB. Then pick any point C on the circumference and join it to A and B. Triangle ABC appeared.

Now measure the angle at C. It comes out 90°. You move C somewhere else on the circumference? 90° again. Anywhere. Every time a right angle.

This is Thales’ theorem: every angle on the diameter of a circumference, with its vertex on the circumference, is right. Thales of Miletus discovered it 2600 years ago — even before Pythagoras lived.

It sounds like magic, but it has a reason you will see today. And most of all it has a use: Thales’ circle is a machine for right angles for free. You need to draw a right-angled triangle with a given hypotenuse? A tangent to a circumference? Check that a table top is right-angled? Thales will arrange it without a protractor.

Get out the compasses — today you draw and wonder.

The class is drawing. The teacher set: a right-angled triangle with hypotenuse 8 cm. Adam wrestles with a protractor and every time he gets 88° or 92°. Tereza goes differently: she draws a line AB 8 cm long, finds the midpoint, takes 4 cm into the compasses and draws a semicircle. Then she PICKS a point C on it — anywhere she likes — and joins. She checks with a protractor: 90.0°. Adam looks: “How come yours comes out?” — “It does not come out. That is Thales. On the diameter there is always a right angle, I do not have to measure it.”

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Thales’ circle is drawn on a DIAMETER: look for the centre in the middle of the line AB and the radius is half of AB. The vertex of the right angle then lies anywhere on the circumference — except points A and B.

Why the theorem holds and how to draw with it

Why it works: join point C to the centre S of the circumference. The lines SA, SB and SC are all radii — the same length. Two isosceles triangles appear (ASC and BSC) and when you add their angles, at C they always add up to exactly half of 180° — so 90°. You will see the details at secondary school; now it is enough to know that equal lengths of radii stand behind it, no accident.

Drawing a right-angled triangle with hypotenuse AB:

  1. Draw the line AB (that will be the hypotenuse).
  2. Find its midpoint S — measure and halve.
  3. Draw a circumference with centre S and radius SA (Thales’ circle).
  4. Choose a point C on the circumference and join to A and B. The angle at C is right — guaranteed.

Using backwards — a test for a right angle: you have a triangle and you want to know if the angle at C is right? Draw Thales’ circle on AB. Does C lie on it? Right. Inside the circumference? Obtuse. Outside? Acute.

Thales and Pythagoras complement each other brilliantly: Thales makes the right angle, Pythagoras calculates the sides for it.

Example 1: a triangle with hypotenuse 10 cm

Task: draw a right-angled triangle ABC with hypotenuse AB = 10 cm and shorter side AC = 6 cm.

Method: AB = 10 cm, midpoint S, Thales’ circle with radius 5 cm.

Now the second condition: AC = 6 cm. I take 6 cm into the compasses, stick into A and cut Thales’ circle — the intersection is C.

I join and I have a triangle. Check with Pythagoras: BC² = 10² − 6² = 64, BC = 8 cm. I measure on the drawing: 8 cm. Both theorems confirmed each other’s work.

Example 2: is the corner really right?

Filip cut a triangular shelf from wood and claims it has a right angle. Sides: 12 cm, 16 cm and hypotenuse 20 cm.

A Thales test would want drawing. Faster here is Pythagoras: 12² + 16² = 144 + 256 = 400 = 20². Right angle confirmed.

And when is Thales useful? When you need to DRAW an angle or when you have not measured all the sides — a hypotenuse and compasses are enough. Two theorems, two tools; pick according to the situation.

Example 3: where can an observer stand

A straight gallery wall has a painting on it from point A to point B. At what angle do you see the painting?

Stand so that you see the painting at a right angle — where all can that be?

Thales knows the answer: exactly on the circumference on diameter AB. إغلاقr to the wall (inside the circumference) you see the painting at an obtuse, wider angle; further (outside) at a sharper one.

This is not a school toy: camera operators and architects use the same thinking when they deal with viewing angles.

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Thales’ theorem only holds on a DIAMETER. A triangle on an ordinary chord has no right angle. And vertex C must not merge with A or B — there no triangle appears.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. البداية with ten. When it goes well, add more.

Practise Pythagoras’ theorem — with Thales they make a pair: one makes the angle, the other calculates the sides.

Practise the questions

On paper

Compasses, a ruler, a protractor for a check. Draw and convince yourself.

  1. Draw a line AB = 8 cm, its Thales’ circle and three different points C₁, C₂, C₃ on it. انضمام each to A and B and check with a protractor that all three angles are right.

  2. Draw a right-angled triangle with hypotenuse 12 cm and a shorter side 7 cm (Thales’ circle + compasses from point A).

  3. With the triangle from step 2 calculate the other shorter side with Pythagoras and compare with the measured value.

  4. Choose a point C inside Thales’ circle and another outside. Measure the angles at C and write how they change: inside obtuse, outside acute.

Now you 💪

  1. I can draw Thales’ circle on a given hypotenuse.
  2. I know a right angle only appears on a diameter, not on a chord.
  3. I can combine Thales with Pythagoras: draw and calculate.
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Done when: You have a drawn Thales’ circle with three checked right angles and a constructed triangle with a calculated shorter side.

What to take from this lesson

© 2026 Ing. Martin Polak / AlgoRhino · شروط استخدام المحتوى