Viviani's theorem, named after
Vincenzo Viviani
Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo. , states that the sum of the distances from ''any'' interior point to the sides of an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
equals the length of the triangle's
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
.
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
![Viviani_theorem_visual_proof](https://upload.wikimedia.org/wikipedia/commons/4/44/Viviani_theorem_visual_proof.svg)
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.
[Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA 2010, , p. 96 () ]
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
,
, and
. 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:
:
and thus
:
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 plot
A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle 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 equi ...
s, such as
flammability diagram
Flammability diagrams show the control 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, known as a ternary plot. Such diagrams are availab ...
s.
More generally, they allow one to give coordinates on a regular
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
in the same way.
Extensions
Parallelogram
The sum of the distances from any interior point of a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
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
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.
[
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 (both equiangular and equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
), 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
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
, 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
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
.[
]
Equiangular polygon
The sum of the distances from an interior point to the sides of an 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 (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which ...
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
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
.[
]
References
Further reading
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External links
*
Li Zhou, Viviani Polytopes and Fermat Points
* at Cut the knot.
* the Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
* a
Dynamic Geometry Sketches
an interactive dynamic geometry sketch.
*
* {{cite arXiv, first1=Addie , last1=Armstrong , first2=Dan , last2=McQuillan , title=Specialization, generalization, and a new proof of Viviani's theorem, eprint=1701.01344, year=2017, class=math.HO
Theorems about equilateral triangles
Articles containing proofs