In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by
d
2
=
R
(
R
−
2
r
)
{\displaystyle d^{2}=R(R-2r)}
or equivalently
1
R
−
d
+
1
R
+
d
=
1
r
,
{\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}
where
R
{\displaystyle R}
and
r
{\displaystyle r}
denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, 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
≥
2
r
,
{\displaystyle R\geq 2r,}
which holds with equality only in the equilateral case.
Contents
Proof
Let
O
{\displaystyle O}
be the center of the circumcircle of triangle
A
B
C
{\displaystyle ABC}
, and let
I
{\displaystyle I}
be the center of its incircle.
If the ray
A
I
{\displaystyle AI}
intersects the circumcircle at the point
L
{\displaystyle L}
, then
L
{\displaystyle L}
is the midpoint of arc
B
C
{\displaystyle BC}
. Draw the ray
L
O
{\displaystyle LO}
and denote its intersection with the circumcircle by
Stronger version of the inequality
A stronger version is
R
r
≥
a
b
c
+
a
3
+
b
3
+
c
3
2
a
b
c
≥
a
b
+
b
c
+
c
a
−
1
≥
2
3
(
a
b
Euler's theorem for the escribed circle
If
r
a
{\displaystyle r_{a}}
and
d
a
{\displaystyle d_{a}}
denote respectively the radius of the escribed circle opposite to the vertex
A
{\displaystyle A}
and the distance between its center and the center of
the circumscribed circle, then
d
a
2
=
R
(
R
+
2
r
a
)
{\displaystyle 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.