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[[File:Satzviviani.PNG|thumb|right|250px|The sum {{nowrap|''s'' + ''u'' + ''t''}} of the lengths is the height of the triangle.]]
'''Viviani's theorem''', named after [[Vincenzo Viviani]], states that the sum of the distances from ''any'' interior point to the sides of an [[equilateral triangle]] equals the length of the triangle's [[Altitude (triangle)|altitude]].<ref name="ref1">[http://arxiv.org/abs/0903.0753v3 Elias Abboud "On Viviani’s Theorem and its Extensions"] pp. 2, 11</ref>

==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 ''u, s, t'' 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 <math>\frac{u \cdot a}{2}</math>, <math>\frac{s \cdot a}{2}</math>, and <math>\frac{t \cdot a}{2}</math>.  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:

:<math>\frac{u \cdot a}{2} + \frac{s \cdot a}{2} + \frac{t \cdot a}{2} = \frac{h \cdot a}{2}</math>

and thus

:''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.<ref name=Chen>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>

== Applications ==
[[Image:Flammability diagram methane.png|right|thumb|400px|[[Flammability diagram]] for methane]]
{{details|Ternary plot}}

Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making [[ternary plot]]s, such as [[flammability diagram]]s.

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.<ref name=Chen/>

The result generalizes to any 2''n''-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 polygon|regular]] (both equiangular and [[equilateral polygon|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.<ref name=Chen/><ref>Pickover, Clifford A., ''The Math Book'', Stirling, 2009: p. 150.</ref> However, the converse does not hold; the non-square parallelogram is a counterexample.<ref name=Chen/>

===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.<ref name="ref1" /> 

===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]].<ref name=Chen/>

==References==
{{reflist}}
* {{MathWorld|title=Viviani's Theorem|urlname=VivianisTheorem}}

*[http://arxiv.org/abs/1008.1236 Li Zhou, Viviani Polytopes and Fermat Points]

==External links==
*[http://www.cut-the-knot.org/Curriculum/Geometry/Viviani.shtml Viviani's Theorem: What is it?] at [[Cut the knot]].
*[http://demonstrations.wolfram.com/VivianisTheorem/ Viviani's Theorem] by Jay Warendorff, the [[Wolfram Demonstrations Project]].
* [http://frink.machighway.com/~dynamicm/viviani-general.html Some generalizations of Viviani's theorem] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], an interactive dynamic geometry sketch.
* [http://frink.machighway.com/~dynamicm/clough.html Clough's Theorem - a variation of Viviani's theorem and some generalizations] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], an interactive dynamic geometry sketch.

[[Category:Polygons]]
[[Category:Triangle geometry]]
[[Category:Theorems in geometry]]
[[Category:Articles containing proofs]]