science

Lemoine's problem

Lemoine's problem is a science 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 Lemoine's problem rather than just read about it. In short: In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868: Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle. The problem was published as Question 864 in Nouvelles Annales de Mathématiques (Series 2, Volume 7 (1868), p 191).

Key takeaways

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

Reference excerpt

In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868:

Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle. The problem was published as Question 864 in Nouvelles Annales de Mathématiques (Series 2, Volume 7 (1868), p 191). The chief interest in the problem is that a discussion of the solution of the problem by Ludwig Kiepert published in Nouvelles Annales de Mathématiques (series 2, Volume 8 (1869), pp 40–42) contained a description of a hyperbola which is now known as the Kiepert hyperbola.

Ludwig Kiepert's solution

Kiepert establishes the validity of his construction by proving a few lemmas.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Lemoine's problem

Start with the simplest possible case. Write down what Lemoine's problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Lemoine's problem 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 Lemoine's problem 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 Lemoine's problem

In research: Lemoine's problem appears in science 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 Lemoine's problem 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: Lemoine's problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Triangle problems, so understanding it makes those chapters shorter.
In everyday life: Look for Lemoine's problem 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 Lemoine's problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lemoine's problem 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 Lemoine's problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lemoine's problem in simple terms?

In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868: Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle. The problem was published as Question 864…

Why does Lemoine's problem matter?

Because it connects several science 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 Lemoine's problem?

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 Lemoine's problem.