Euler's theorem in geometry
- Euler's theorem in geometry
In geometry, Euler's theorem, named after Leonhard Euler, states that the distance "d" between the circumcentre and incentre of a triangle can be expressed as
:
where "R" and "r" denote the circumradius and inradius respectively (the radii of the above two circles).
From the theorem follows the Euler inequality::
Proof
Let "O" be the circumcentre of triangle "ABC", and "I" be its incentre, the extension of "AI" intersects the circumcircle at "L", then "L" is the mid-point of arc "BC". Join "LO" and extend it so that it intersects the circumcircle at "M". From "I" construct a perpendicular to AB, and let D be its foot, then "ID" = "r". It is not difficult to prove that triangle "ADI" is similar to triangle "MBL", so "ID" / "BL" = "AI" / "ML", i.e. "ID" × "ML" = "AI" × "BL". Therefore 2"Rr" = "AI" × "BL". Join "BI", because
: angle "BIL" = angle "A" / 2 + angle "ABC" / 2,
: angle "IBL" = angle "ABC" / 2 + angle "CBL" = angle "ABC" / 2 + angle "A" / 2,
therefore angle "BIL" = angle "IBL", so "BL" = "IL", and "AI" × "IL" = 2"Rr". Extend "OI" so that it intersects the circumcircle at "P" and "Q", then "PI" × "QI" = "AI" × "IL" = 2"Rr", so ("R" + "d")("R" − "d") = 2"Rr", i.e. "d"2 = "R"("R" − 2"r").
External links
* [http://mathworld.wolfram.com/EulerTriangleFormula.html Euler's theorem on MathWorld]
Wikimedia Foundation.
2010.
Look at other dictionaries:
Euler's theorem (differential geometry) — In the mathematical field of differential geometry, Euler s theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions… … Wikipedia
Euler's formula — This article is about Euler s formula in complex analysis. For Euler s formula in algebraic topology and polyhedral combinatorics see Euler characteristic. Part of a series of articles on The mathematical constant e … Wikipedia
List of topics named after Leonhard Euler — In mathematics and physics, there are a large number of topics named in honour of Leonhard Euler (pronounced Oiler ). As well, many of these topics include their own unique function, equation, formula, identity, number (single or sequence), or… … Wikipedia
Geometry — (Greek γεωμετρία ; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of… … Wikipedia
Theorem — The Pythagorean theorem has at least 370 known proofs[1] In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements … Wikipedia
Euler's identity — [ 300px|thumb|right|The exponential function e z can be defined as the limit of nowrap|(1 + z / N ) N , as N approaches infinity, and thus e iπ is the limit of nowrap|(1 + iπ/N ) N . In this animation N takes various increasing values from 1 to… … Wikipedia
Euler system — In mathematics, an Euler system is a technical device in the theory of Galois modules, first noticed as such in the work around 1990 by Victor Kolyvagin on Heegner points on modular elliptic curves. This concept has since undergone an axiomatic… … Wikipedia
Leonhard Euler — Infobox Scientist name = Leonhard Euler|box width = 300px |200px image width = 200px caption = Portrait by Johann Georg Brucker birth date = birth date|df=yes|1707|4|15 birth place = Basel, Switzerland death date = 18 September (O.S 7 September)… … Wikipedia
List of geometry topics — This is list of geometry topics, by Wikipedia page.*Geometric shape covers standard terms for plane shapes *List of mathematical shapes covers all dimensions *List of differential geometry topics *List of geometers *See also list of curves, list… … Wikipedia
Pappus's theorem — ▪ geometry in mathematics, theorem named for the 4th century Greek geometer Pappus of Alexandria that describes the volume of a solid, obtained by revolving a plane region D about a line L not intersecting D, as the product of the area of D and… … Universalium
