mathematics

Pompeiu's theorem

Pompeiu'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 Pompeiu's theorem rather than just read about it. In short: Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical.

Key takeaways

Pompeiu'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 Pompeiu's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pompeiu's theorem from memory before moving on to harder problems.

Reference excerpt

Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following:

Given an equilateral triangle ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a (maybe, degenerate) triangle.

The proof is quick. Consider a rotation of 60° about the point B. Assume A maps to C, and P maps to P′. Then PB = P′B, and ∠PBP′ = 60°. Hence triangle PBP′ is equilateral and PP′ = PB. Then PA = P′C. Thus, triangle PCP′ has sides equal to PA, PB, and PC and the proof by construction is complete (see drawing). Further investigations reveal that if P is not in the interior of the triangle, but rather on the circumcircle, then PA, PB, PC form a degenerate triangle, with the largest being equal to the sum of the others; this observation is also known as Van Schooten's theorem. Generally, by the point P and the lengths to the vertices of the equilateral triangle – PA, PB, and PC two equilateral triangles (the larger and the smaller) with sides a1 and a2 are defined:

a 1 , 2 2 = 1 2 ( P A 2 + P B 2 + P C 2 ± 4 3 △ ( P A , P B , P C ) ) . {\displaystyle a_{1,2}^{2}={\frac {1}{2}}\left(PA^{2}+PB^{2}+PC^{2}\pm 4{\sqrt {3}}\triangle _{(PA,PB,PC)}\right).}

The symbol △ denotes the area of the triangle whose sides have lengths PA, PB, PC. Pompeiu published the theorem in 1936; however August Ferdinand Möbius had already published a more general theorem about four points in the Euclidean plane in 1852. In this paper Möbius also derived the statement of Pompeiu's theorem explicitly as a special case of his more general theorem. For this reason, the theorem is also known as the Möbius–Pompeiu theorem.

External links MathWorld's page on Pompeiu's Theorem Pompeiu's theorem at cut-the-knot.org

Notes

Illustrations

Worked examples

Example 1 — a first encounter with Pompeiu's theorem

Start with the simplest possible case. Write down what Pompeiu'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 Pompeiu'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 Pompeiu'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 Pompeiu's theorem

In research: Pompeiu'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 Pompeiu'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: Pompeiu's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, Theorems about equilateral triangles, Theorems about triangles and circles, so understanding it makes those chapters shorter.
In everyday life: Look for Pompeiu'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 Pompeiu's theorem in 20 minutes

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

Frequently asked questions

What is Pompeiu's theorem in simple terms?

Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical.

Why does Pompeiu'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 Pompeiu'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 Pompeiu's theorem.