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Euler's theorem in geometry

{{short description|On distance between centers of a triangle}}

File:Euler theorem2.svg

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by{{r|Johnson|Dunham}}

d^2=R (R-2r)

or equivalently

\frac{1}{R-d} + \frac{1}{R+d} = \frac{1}{r},

where R and 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.{{r|LevershaSmith}} However, the same result was published earlier by William Chapple in 1746.{{r|Chapple}}

From the theorem follows the Euler inequality:{{r|wlim}}

R \ge 2r,

which holds with equality only in the equilateral case.{{r|SV}}

Stronger version of the inequality

A stronger version{{r|SV}} is

\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, b, and c are the side lengths of the triangle.

Euler's theorem for the excribed circle

If 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.{{r|PS}}

See also

References

{{reflist|refs=

{{citation

| last = Chapple | first = William | author-link = William Chapple (surveyor)

| journal = Miscellanea Curiosa Mathematica

| pages = 117–124

| title = An essay on the properties of triangles inscribed in and circumscribed about two given circles

| url = https://archive.org/details/miscellaneacuri01unkngoog/page/n142

| volume = 4

| year = 1746}}. The formula for the distance is near the bottom of p.123.

{{citation

| last = Dunham | first = William

| isbn = 9780883855584

| page = 300

| publisher = Mathematical Association of America

| series = Spectrum Series

| title = The Genius of Euler: Reflections on his Life and Work

| url = https://books.google.com/books?id=M4-zUnrSxNoC&pg=PA300

| volume = 2

| year = 2007}}

{{citation|last=Johnson|first=Roger A.|title=Advanced Euclidean Geometry|publisher=Dover Publ.|year=2007|orig-year=1929|page=186}}

{{citation

| last1 = Leversha | first1 = Gerry

| last2 = Smith | first2 = G. C.

| date = November 2007

| issue = 522

| journal = The Mathematical Gazette

| jstor = 40378417

| pages = 436–452

| title = Euler and triangle geometry

| volume = 91| doi = 10.1017/S0025557200182087

| s2cid = 125341434

}}

{{citation

| last1 = Alsina | first1 = Claudi

| last2 = Nelsen | first2 = Roger

| isbn = 9780883853429

| page = 56

| publisher = Mathematical Association of America

| series = Dolciani Mathematical Expositions

| title = When Less is More: Visualizing Basic Inequalities

| url = https://books.google.com/books?id=U1ovBsSRNscC&pg=PA56

| volume = 36

| year = 2009}}

{{citation

| last1 = Svrtan | first1 = Dragutin

| last2 = Veljan | first2 = Darko

| journal = Forum Geometricorum

| pages = 197–209

| title = Non-Euclidean versions of some classical triangle inequalities

| url = https://forumgeom.fau.edu/FG2012volume12/FG201217index.html

| volume = 12

| year = 2012}}; see p. 198

{{citation

| last1 = Pambuccian | first1 = Victor

| last2 = Schacht | first2 = Celia

| doi = 10.1007/s00022-018-0414-6

| journal = Journal of Geometry

| pages = 1–11

| title = Euler's inequality in absolute geometry

| volume = 109 (Art. 8)

| year = 2018| s2cid = 125459983

}}

}}

External links

{{Commons category|Euler's theorem in geometry}}

  • {{mathworld|id=EulerTriangleFormula|title=Euler Triangle Formula|mode=cs2}}

Category:Articles containing proofs

Category:Triangle inequalities

Category:Theorems about triangles and circles