Viviani's theorem, named after Vincenzo Viviani

Vincenzo Viviani

Vincenzo Viviani was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo.-Biography:...

, states that the sum of the distances from a point to the sides of an equilateral triangle equals the length of the triangle's altitude

Altitude (triangle)

In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...

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The theorem can be extended to equilateral polygons and equiangular polygon

Equiangular polygon

In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon.The only equiangular triangle is the equilateral triangle...

s. Specifically, the sum of distances from a point to the side lines of an equiangular (or equilateral) polygon does not depend on the point.

Proof

This theorem can be proven by comparing areas of triangles. Let ABC be an equilateral triangle where h is the height, and s is the length of each side. P is any point inside the triangle, and ℓ, m, n are the distances of point P from the sides. Then the area of triangle ABC is:

Applications

Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plot

Ternary plot

A ternary plot, ternary graph, triangle plot, simplex plot, or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle...

s, such as flammability diagram

Flammability diagram

Flammability diagrams show the regimes of flammability in mixtures of fuel, oxygen and an inert gas, typically nitrogen. Mixtures of the three gasses are usually depicted in a triangular diagram, also known as a Ternary plot. Such diagrams are available in the speciality literature...

s.

More generally, they allow one to give coordinates on a regular simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 in the same way.

External links

• Viviani's Theorem: What is it? at Cut the knot.
• Viviani's Theorem by Jay Warendorff, the Wolfram Demonstrations Project
  Wolfram Demonstrations Project
  The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
  
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