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Thales's theorem

Thales's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Thales's theorem rather than just read about it. In short: In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements.

Thales's theorem — main illustration
Thales's theorem — illustration

Key takeaways

  • Thales's theorem belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Thales's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Thales's theorem from memory before moving on to harder problems.

Reference excerpt

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

History

Thales of Miletus (early 6th century BC) is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) documenting Pamphila's (1st century AD) statement that Thales "was the first to inscribe in a circle a right-angle triangle". Thales was claimed to have traveled to Egypt and Babylonia, where he is supposed to have learned about geometry and astronomy and thence brought their knowledge to the Greeks, along the way inventing the concept of geometric proof and proving various geometric theorems. However, there is no direct evidence for any of these claims, and they were most likely invented speculative rationalizations. Modern scholars believe that Greek deductive geometry as found in Euclid's Elements was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational. The theorem appears in Book III of Euclid's Elements (c. 300 BC) as proposition 31: "In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle." Dante Alighieri's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.

Proof

First proof The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal.

Since OA = OB = OC, △OBA and △OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠ OBC = ∠ OCB and ∠ OBA = ∠ OAB. Let α = ∠ BAO and β = ∠ OBC. The three internal angles of the ∆ABC triangle are α, (α + β), and β. Since the sum of the angles of a triangle is equal to 180°, we have

α + ( α + β ) + β = 180 ∘ 2 α + 2 β = 180 ∘ 2 ( α + β ) = 180 ∘ ∴ α + β = 90 ∘ . {\displaystyle {\begin{aligned}\alpha +(\alpha +\beta )+\beta &=180^{\circ }\\2\alpha +2\beta &=180^{\circ }\\2(\alpha +\beta )&=180^{\circ }\\\therefore \alpha +\beta &=90^{\circ }.\end{aligned}}}

Q.E.D.

Second proof The theorem may also be proven using trigonometry: Let O = (0, 0), A = (−1, 0), and C = (1, 0). Then B is a point on the unit circle (cos θ, sin θ). We will show that △ABC forms a right angle by proving that AB and BC are perpendicular — that is, the product of their slopes is equal to −1. We calculate the slopes for AB and BC:

… excerpt ends here. Continue reading the full article.

Illustrations

Thales's theorem: Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle.
Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle.
Thales's theorem: Non si est dare primum motum esse o se del mezzo cerchio far si puote triangol sì c'un recto nonauesse. – Dante's Paradiso, Canto 13, lines 100–102 Non si est dare primum motum esse, Or if in semicircle can be made Triangle so that it have no right angle. – English translation by Longfellow
Non si est dare primum motum esse o se del mezzo cerchio far si puote triangol sì c'un recto nonauesse. – Dante's Paradiso, Canto 13, lines 100–102 Non si est dare primum motum esse, Or if in semicircle can be made Triangle so that it have no right angle. – English translation by Longfellow
Thales's theorem illustration
Thales's theorem illustration
Thales's theorem: Showing a proof of Thales' theorem by rotating the triangle and observing that you get a rectangle in this way (both diagonals are the same length)
Showing a proof of Thales' theorem by rotating the triangle and observing that you get a rectangle in this way (both diagonals are the same length)

Worked examples

Example 1 — a first encounter with Thales's theorem

Start with the simplest possible case. Write down what Thales's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Thales's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Thales's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Thales's theorem

In research
Thales's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Thales's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Thales's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ancient Greek mathematics, Euclidean plane geometry, Theorems about right triangles, so understanding it makes those chapters shorter.
In everyday life
Look for Thales's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Thales's theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Thales's theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Thales's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Thales's theorem in simple terms?

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book…

Why does Thales's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Thales's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Thales's theorem.

Tags

  • Ancient Greek mathematics
  • Euclidean plane geometry
  • Theorems about right triangles
  • Theorems about triangles and circles

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