geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

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 bis ...

of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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, a circumscribed circle for a set of points is a circle passing through each of them. Such a circle is said to ''circumscribe'' the points or a polygon formed from them; such a polygon is said to be ''inscribed'' in the circle. * Circu ...

and

inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incente ...

respectively). The theorem is named for

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

, 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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

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 excribed 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. The term was introduced by ...

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 excribed circle

Euler's inequality in absolute geometry

See also

References

External links