Menelaus's theorem

Key takeaways

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

Reference excerpt

In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that

| A F ¯ F B ¯ | × | B D ¯ D C ¯ | × | C E ¯ E A ¯ | = 1 , {\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=1,}

where "| |" denotes absolute value (i.e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, A F ¯ F B ¯ {\displaystyle {\tfrac {\overline {AF}}{\overline {FB}}}} is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states

A F ¯ F B ¯ × B D ¯ D C ¯ × C E ¯ E A ¯ = − 1. {\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1.}

Equivalently,

A F ¯ × B D ¯ × C E ¯ = − F B ¯ × D C ¯ × E A ¯ . {\displaystyle {\overline {AF}}\times {\overline {BD}}\times {\overline {CE}}=-{\overline {FB}}\times {\overline {DC}}\times {\overline {EA}}.}

Some authors organize the factors differently and obtain the seemingly different relation

F A ¯ F B ¯ × D B ¯ D C ¯ × E C ¯ E A ¯ = 1 , {\displaystyle {\frac {\overline {FA}}{\overline {FB}}}\times {\frac {\overline {DB}}{\overline {DC}}}\times {\frac {\overline {EC}}{\overline {EA}}}=1,}

but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that

… excerpt ends here. Continue reading the full article.

Illustrations

Menelaus's theorem: Homothetie centers D, E F are colinear iff the composition is identity. — Homothetie centers D, E F are colinear iff the composition is identity.

Worked examples

Example 1 — a first encounter with Menelaus's theorem

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

In research: Menelaus'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 Menelaus'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: Menelaus's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Affine geometry, Ancient Greek mathematics, Euclidean plane geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Menelaus'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.

How to study Menelaus's theorem in 20 minutes

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

Frequently asked questions

What is Menelaus's theorem in simple terms?

In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C.

Why does Menelaus'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 Menelaus'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 Menelaus's theorem.