Viviani's theorem

The sum ℓ + m + n of the lengths is the height of the triangle.

Viviani's theorem, named after Vincenzo Viviani, 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.

The theorem can be extended to equilateral polygons and equiangular polygons. 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 proof depends on the readily-proved proposition that the area of a triangle is half its base times its height.

Let ABC be an equilateral triangle whose height is h and whose side is s.

Let P be any point inside the triangle, and x, y, z the 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 s*x/2, s*y/2, and s*z/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:

s*x/2 + s*y/2 + s*z/2 = s*h/2

and thus

x + y + z = h

Q E D

Applications

Flammability diagram for methane

Further information: Ternary plot

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.

See also

• Ternary plot

References

• Weisstein, Eric W. "Viviani's Theorem". MathWorld.

• Li Zhou, Viviani Polytopes and Fermat Points

External links

• Viviani's Theorem: What is it? at Cut the knot.
• Viviani's Theorem by Jay Warendorff, the Wolfram Demonstrations Project.
• Some generailizations of Viviani's theorem at Dynamic Geometry Sketches, an interactive dynamic geometry sketch.