geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Euler's theorem states that the distance ''d'' between the

circumcenter 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 ...

and

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bise ...

of a

triangle 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- colli ...

is given by

d^2=R (R-2r)

or equivalently

\frac + \frac = \frac,

where

R

and

r

denote the circumradius and inradius respectively (the radii of the

circumscribed circle 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 poly ...

and inscribed circle respectively). The theorem is named for

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

, who published it in 1765. However, the same result was published earlier by William Chapple in 1746. From the theorem follows the Euler inequality:

R \ge 2r,

which holds with equality only in the

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 oth ...

case.

Stronger version of the inequality

A stronger version is

\frac \geq \frac \geq \frac+\frac+\frac-1 \geq \frac \left(\frac+\frac+\frac \right) \geq 2,

where

a

b

, and

c

are the side lengths of the triangle.

Euler's theorem for the escribed circle

r_a

and

d_a

denote respectively the radius of the escribed circle opposite to the vertex

A

and the distance between its center and the center of the circumscribed circle, then

d_a^2=R(R+2r_a)

Euler's inequality in absolute geometry

Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in

absolute geometry Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not su ...

References

External links

*{{mathworld, id=EulerTriangleFormula, title=Euler Triangle Formula, mode=cs2 Articles containing proofs Triangle inequalities Theorems about triangles and circles

Stronger version of the inequality

Euler's theorem for the escribed circle

Euler's inequality in absolute geometry

See also

References

External links