mathematics

Routh's theorem

Routh'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 Routh's theorem rather than just read about it. In short: In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle A B C {\displaystyle ABC} points D {\displaystyle D} , E {\displaystyle E} , and F {\displaystyle F} lie on segments B C {\displaystyle BC} , C A {\displaystyle CA} , and A B {\displaystyle AB} , then writing C D B D = x {\dis…

Key takeaways

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

Reference excerpt

In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle A B C {\displaystyle ABC} points D {\displaystyle D} , E {\displaystyle E} , and F {\displaystyle F} lie on segments B C {\displaystyle BC} , C A {\displaystyle CA} , and A B {\displaystyle AB} , then writing C D B D = x {\displaystyle {\tfrac {CD}{BD}}=x} , A E C E = y {\displaystyle {\tfrac {AE}{CE}}=y} , and B F A F = z {\displaystyle {\tfrac {BF}{AF}}=z} , the signed area of the triangle formed by the cevians A D {\displaystyle AD} , B E {\displaystyle BE} , and C F {\displaystyle CF} is

S A B C ⋅ ( x y z − 1 ) 2 ( x y + y + 1 ) ( y z + z + 1 ) ( z x + x + 1 ) , {\displaystyle S_{ABC}\cdot {\frac {(xyz-1)^{2}}{(xy+y+1)(yz+z+1)(zx+x+1)}},}

where S A B C {\displaystyle S_{ABC}} is the area of the triangle A B C {\displaystyle ABC} . This theorem was given by Edward John Routh on page 82 of his Treatise on Analytical Statics with Numerous Examples in 1896. The particular case x = y = z = 2 {\displaystyle x=y=z=2} has become popularized as the one-seventh area triangle. The x = y = z = 1 {\displaystyle x=y=z=1} case implies that the three medians are concurrent (through the centroid).

Proof

Suppose that the area of triangle A B C {\displaystyle ABC} is 1. For triangle A B D {\displaystyle ABD} and line F R C {\displaystyle FRC} , Menelaus's theorem implies

A F F B × B C C D × D R R A = − 1 {\displaystyle {\frac {AF}{FB}}\times {\frac {BC}{CD}}\times {\frac {DR}{RA}}=-1} . Then D R R A = B F F A × C D C B = z x x + 1 {\displaystyle {\frac {DR}{RA}}={\frac {BF}{FA}}\times {\frac {CD}{CB}}={\frac {zx}{x+1}}} . Thus the area of triangle A R C {\displaystyle ARC} is

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Routh's theorem

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

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

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

Frequently asked questions

What is Routh's theorem in simple terms?

Why does Routh'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 Routh'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 Routh's theorem.