Viviani's theorem

Key takeaways

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

Reference excerpt

Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.

Proof

This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side. Let ABC be an equilateral triangle whose height is h and whose side is a. Let P be any point inside the triangle, and s, t, u the perpendicular distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA. Now, the areas of these triangles are u ⋅ a 2 {\displaystyle {\frac {u\cdot a}{2}}} , s ⋅ a 2 {\displaystyle {\frac {s\cdot a}{2}}} , and t ⋅ a 2 {\displaystyle {\frac {t\cdot a}{2}}} . They exactly fill the enclosing triangle, so the sum of these areas is equal to the area of the enclosing triangle. So we can write:

u ⋅ a 2 + s ⋅ a 2 + t ⋅ a 2 = h ⋅ a 2 {\displaystyle {\frac {u\cdot a}{2}}+{\frac {s\cdot a}{2}}+{\frac {t\cdot a}{2}}={\frac {h\cdot a}{2}}}

and thus

u + s + t = h {\displaystyle u+s+t=h}

Q.E.D.

Converse The converse also holds: If the sum of the distances from an interior point of a triangle to the sides is independent of the location of the point, the triangle is equilateral.

Applications

Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plots, such as flammability diagrams. More generally, they allow one to give coordinates on a regular simplex in the same way.

Extensions

Parallelogram The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram. The result generalizes to any 2n-gon with opposite sides parallel. Since the sum of distances between any pair of opposite parallel sides is constant, it follows that the sum of all pairwise sums between the pairs of parallel sides, is also constant. The converse in general is not true, as the result holds for an equilateral hexagon, which does not necessarily have opposite sides parallel.

Regular polygon If a polygon is regular (both equiangular and equilateral), the sum of the distances to the sides from an interior point is independent of the location of the point. Specifically, it equals n times the apothem, where n is the number of sides and the apothem is the distance from the center to a side. However, the converse does not hold; the non-square parallelogram is a counterexample.

Equilateral polygon The area proof given above for the equilateral triangle generalizes to any (convex) equilateral polygon. In other words, if a polygon has all sides equal, the sum of the distances from an interior point to the sides is independent of the location of the point. The result follows as before, since the equilateral n-gon can be divided up into n triangles all with equal base, say a, but with different heights. But the sum of all the areas of these triangles equal the area A of the polygon, so as before, A = 1/2 a (sum of the distances to the sides).

Equiangular polygon The sum of the distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point.

Convex polygon A necessary and sufficient condition for a convex polygon to have a constant sum of distances from any interior point to the sides is that there exist three non-collinear interior points with equal sums of distances.

Regular polyhedron The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra.

References

Further reading Alsina, Claudi; Nelsen, Roger B. (2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. Vol. 50. Mathematical Association of America. ISBN 978-1-61444-216-5. Gueron, Shay; Tessler, Ran (2002). "The Fermat-Steiner problem". Amer. Math. Monthly. 109 (5): 443–451. doi:10.2307/2695644. JSTOR 2695644. Samelson, Hans (2003). "Proof without words: Viviani's theorem with vectors". Math. Mag. 76 (3): 225. doi:10.2307/3219327. JSTOR 3219327. Chen, Zhibo; Liang, Tian (2006). "The converse of Viviani's theorem". The College Mathematics Journal. 37 (5): 390–391. doi:10.2307/27646392. JSTOR 27646392. Kawasaki, Ken-Ichiroh; Yagi, Yoshihiro; Yanagawa, Katsuya (2005). "On Viviani's theorem in three dimensions". Math. Gaz. 89 (515): 283–287. doi:10.1017/S002555720017785X. JSTOR 3621243. S2CID 126113074. Zhou, Li (2012). "Viviani polytopes and Fermat Points". Coll. Math. J. 43 (4): 309–312. arXiv:1008.1236. CiteSeerX 10.1.1.740.7670. doi:10.4169/college.math.j.43.4.309. S2CID 117039483. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)

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Illustrations

Worked examples

Example 1 — a first encounter with Viviani's theorem

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

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

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

Frequently asked questions

What is Viviani's theorem in simple terms?

Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics e…

Why does Viviani'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 Viviani'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 Viviani's theorem.