Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of

triangle geometry A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

. It is also referred to as Sylvester's (triangle) problem in literature, when it is given as a problem rather than a theorem. The theorem is named after the British mathematician

James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...

Theorem

Consider three pairwise distinct vectors of equal length

\vec

\vec

and

\vec

each of them acting on the same point

O

thus creating the points

A

B

and

C

. Those points form the triangle

\triangle ABC

with

O

as the center of its

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

. Now let

H

denote the

orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

of the triangle, then connection vector

\overrightarrow

is equal to the sum of the three vectors: :

\overrightarrow=\vec+\vec+\vec

Furthermore, since the points

O

and

H

are located on the

Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, inclu ...

together with the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...

S

the following equation holds:Michael de Villiers: "'Generalising a problem of Sylvester". In: ''The Mathematical Gazette'', volume 96, no. 535 (March 2012), pp 78-81
JSTOR
:

\overrightarrow=3\cdot \overrightarrow

Generalisation

If the condition of equal length in Sylvester's theorem is dropped and one considers merely three arbitrary pairwise distinct vectors, then the equation above does not hold anymore. However the relation with the centroid remains true, that is: :

3\cdot \overrightarrow=\vec+\vec+\vec

This follows directly from the definition of the centroid for a finite set of points in

\mathbb^n

, which also yields a version for

n

vectors acting on

O

: :

n\cdot \overrightarrow=\sum_^n v_i

Here

S

is the centroid of the vertices of the polygon generated by the

n

vectors acting on

O

.Note that the (area) centroid of a polygon with ''n'' vertices differs from the centroid of its vertices for ''n''>3

References

External links

* {{MathWorld , id=/SylvestersTriangleProblem, title=Sylvester's Triangle Problem *Darij Grinberg
''Solution to American Mathematical Monthly Problem 11398 by Stanley Huang''
– contains Sylvester's theorem including its proof as a lemma Theorems about triangles Triangle problems