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On distance between centers of a triangle
Euler's theorem:
d
=
|
I
O
|
=
R
(
R
−
2
r
)
{\displaystyle d=|IO|={\sqrt {R(R-2r)}}}
In
geometry , Euler's theorem states that the distance d between the
circumcenter and
incenter of a
triangle is given by
[1]
[2]
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.
[3] However, the same result was published earlier by
William Chapple in 1746.
[4]
From the theorem follows the Euler inequality :
[5]
R
≥
2
r
,
{\displaystyle R\geq 2r,}
which holds with equality only in the
equilateral case.
[6]
Stronger version of the inequality
A stronger version
[6] 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
+
b
c
+
c
a
)
≥
2
,
{\displaystyle {\frac {R}{r}}\geq {\frac {abc+a^{3}+b^{3}+c^{3}}{2abc}}\geq {\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}-1\geq {\frac {2}{3}}\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2,}
where
a
{\displaystyle a}
,
b
{\displaystyle b}
, and
c
{\displaystyle c}
are the side lengths of the triangle.
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 .
[7]
See also
References
^ Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry , Dover Publ., p. 186
^ Dunham, William (2007),
The Genius of Euler: Reflections on his Life and Work , Spectrum Series, vol. 2, Mathematical Association of America, p. 300,
ISBN
9780883855584
^ Leversha, Gerry; Smith, G. C. (November 2007), "Euler and triangle geometry",
The Mathematical Gazette , 91 (522): 436–452,
doi :
10.1017/S0025557200182087 ,
JSTOR
40378417 ,
S2CID
125341434
^
Chapple, William (1746),
"An essay on the properties of triangles inscribed in and circumscribed about two given circles" , Miscellanea Curiosa Mathematica , 4 : 117–124 . The formula for the distance is near the bottom of p.123.
^ Alsina, Claudi; Nelsen, Roger (2009),
When Less is More: Visualizing Basic Inequalities , Dolciani Mathematical Expositions, vol. 36, Mathematical Association of America, p. 56,
ISBN
9780883853429
^
a
b Svrtan, Dragutin; Veljan, Darko (2012),
"Non-Euclidean versions of some classical triangle inequalities" ,
Forum Geometricorum , 12 : 197–209 ; see p. 198
^ Pambuccian, Victor; Schacht, Celia (2018), "Euler's inequality in absolute geometry", Journal of Geometry , 109 (Art. 8): 1–11,
doi :
10.1007/s00022-018-0414-6 ,
S2CID
125459983
External links