Sylvester's triangle problem

Key takeaways

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

Reference excerpt

Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred to as Sylvester's (triangle) problem in literature, when it is given as a problem rather than a theorem. The theorem is named after the British mathematician James Joseph Sylvester.

Theorem Consider three pairwise distinct vectors of equal length u → {\displaystyle {\vec {u}}} , v → {\displaystyle {\vec {v}}} and w → {\displaystyle {\vec {w}}} each of them acting on the same point O {\displaystyle O} thus creating the points A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} . Those points form the triangle △ A B C {\displaystyle \triangle ABC} with O {\displaystyle O} as the center of its circumcircle. Now let H {\displaystyle H} denote the orthocenter of the triangle, then connection vector O H → {\displaystyle {\overrightarrow {OH}}} is equal to the sum of the three vectors:

O H → = u → + v → + w → {\displaystyle {\overrightarrow {OH}}={\vec {u}}+{\vec {v}}+{\vec {w}}}

Furthermore, since the points O {\displaystyle O} and H {\displaystyle H} are located on the Euler line together with the centroid S {\displaystyle S} the following equation holds:

O H → = 3 ⋅ O S → {\displaystyle {\overrightarrow {OH}}=3\cdot {\overrightarrow {OS}}}

Generalisation

If the condition of equal length in Sylvester's theorem is dropped and one considers merely three arbitrary pairwise distinct vectors, then the equation above does not hold anymore. However, the relation with the centroid remains true, that is:

3 ⋅ O S → = u → + v → + w → {\displaystyle 3\cdot {\overrightarrow {OS}}={\vec {u}}+{\vec {v}}+{\vec {w}}}

This follows directly from the definition of the centroid for a finite set of points in R n {\displaystyle \mathbb {R} ^{n}} , which also yields a version for n {\displaystyle n} vectors acting on O {\displaystyle O} :

n ⋅ O S → = ∑ i = 1 n v i {\displaystyle n\cdot {\overrightarrow {OS}}=\sum _{i=1}^{n}v_{i}}

Here S {\displaystyle S} is the centroid of the vertices of the polygon generated by the n {\displaystyle n} vectors acting on O {\displaystyle O} .

References

External links Weisstein, Eric W. "Sylvester's Triangle Problem". MathWorld. Darij Grinberg: Solution to American Mathematical Monthly Problem 11398 by Stanley Huang – contains Sylvester's theorem including its proof as a lemma

Illustrations

Worked examples

Example 1 — a first encounter with Sylvester's triangle problem

Start with the simplest possible case. Write down what Sylvester's triangle problem 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 Sylvester's triangle 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 Sylvester's triangle 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 Sylvester's triangle problem

In research: Sylvester's triangle problem 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 Sylvester's triangle 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: Sylvester's triangle problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems about triangles, Triangle problems, so understanding it makes those chapters shorter.
In everyday life: Look for Sylvester's triangle 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.

How to study Sylvester's triangle problem in 20 minutes

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

Frequently asked questions

What is Sylvester's triangle problem in simple terms?

Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred to as Sylvester's (triangle) problem in literature, when it is given as a problem rather than a the…

Why does Sylvester's triangle problem 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 Sylvester's triangle 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 Sylvester's triangle problem.