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In
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 ...
, the Euler line, named after
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 ...
( ), is a line determined from any
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 ...
that is not
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 ...
. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the
orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...
, 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 ...
, the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
, the Exeter point and the center of the
nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each s ...
of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
and the
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
.


Triangle centers on the Euler line


Individual centers

Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
. This property is also true for another
triangle center In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...
, the
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle ...
, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them. Other notable points that lie on the Euler line include the
de Longchamps point In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let th ...
, the Schiffler point, the Exeter point, and the Gossard perspector. However, the
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 ...
generally does not lie on the Euler line; it is on the Euler line only for
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
s, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers. The
tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...
of a reference triangle is tangent to the latter's
circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...
at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle. The center of similitude of the orthic and tangential triangles is also on the Euler line..


Proofs


A vector proof

Let ABC be a triangle. A proof of the fact that 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 ...
O, the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
G and the
orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...
H are collinear relies on free vectors. We start by stating the prerequisites. First, G satisfies the relation :\vec+\vec+\vec=0. This follows from the fact that the absolute barycentric coordinates of G are \frac:\frac:\frac. Further, the problem of SylvesterDörrie, Heinrich, "100 Great Problems of Elementary Mathematics. Their History and Solution". Dover Publications, Inc., New York, 1965, , pages 141 (Euler's Straight Line) and 142 (Problem of Sylvester) reads as :\vec=\vec+\vec+\vec. Now, using the vector addition, we deduce that :\vec=\vec+\vec\,\mboxAGO\mbox,\,\vec=\vec+\vec\,\mboxBGO\mbox,\,\vec=\vec+\vec\,\mboxCGO\mbox. By adding these three relations, term by term, we obtain that :3\cdot\vec=\left(\sum\limits_\vec\right)+\left(\sum\limits_\vec\right)=0-\left(\sum\limits_\vec\right)=-\vec. In conclusion, 3\cdot\vec=\vec, and so the three points O, G and H (in this order) are collinear. In Dörrie's book, the Euler line and the problem of Sylvester are put together into a single proof. However, most of the proofs of the problem of Sylvester rely on the fundamental properties of free vectors, independently of the Euler line.


Properties


Distances between centers

On the Euler line the centroid ''G'' is between the circumcenter ''O'' and the orthocenter ''H'' and is twice as far from the orthocenter as it is from the circumcenter:Altshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. Barnes & Noble 1952). :GH=2GO; :OH=3GO. The segment ''GH'' is a diameter of the orthocentroidal circle. The center ''N'' of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter: :ON = NH, \quad OG =2\cdot GN, \quad NH=3GN. Thus the Euler line could be repositioned on a number line with the circumcenter ''O'' at the location 0, the centroid ''G'' at 2''t'', the nine-point center at 3''t'', and the orthocenter ''H'' at 6''t'' for some scale factor ''t''. Furthermore, the squared distance between the centroid and the circumcenter along the Euler line is less than the squared circumradius ''R''2 by an amount equal to one-ninth the sum of the squares of the side lengths ''a'', ''b'', and ''c'': :GO^2=R^2-\tfrac(a^2+b^2+c^2). In addition, :OH^2=9R^2-(a^2+b^2+c^2); :GH^2=4R^2-\tfrac(a^2+b^2+c^2).


Representation


Equation

Let ''A'', ''B'', ''C'' denote the vertex angles of the reference triangle, and let ''x'' : ''y'' : ''z'' be a variable point in
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
; then an equation for the Euler line is :\sin (2A) \sin(B - C)x + \sin (2B) \sin(C - A)y + \sin (2C) \sin(A - B)z = 0. An equation for the Euler line in barycentric coordinates \alpha :\beta :\gamma is :(\tan C -\tan B)\alpha +(\tan A -\tan C)\beta + (\tan B -\tan A)\gamma =0.


Parametric representation

Another way to represent the Euler line is in terms of a parameter ''t''. Starting with the circumcenter (with trilinear coordinates \cos A : \cos B : \cos C) and the orthocenter (with trilinears \sec A : \sec B : \sec C = \cos B \cos C : \cos C \cos A : \cos A \cos B), every point on the Euler line, except the orthocenter, is given by the trilinear coordinates :\cos A + t \cos B \cos C : \cos B + t \cos C \cos A : \cos C + t \cos A \cos B formed as a
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of the trilinears of these two points, for some ''t''. For example: * 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 ...
has trilinears \cos A:\cos B:\cos C, corresponding to the parameter value t=0. * The
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
has trilinears \cos A + \cos B \cos C : \cos B + \cos C \cos A : \cos C + \cos A \cos B, corresponding to the parameter value t=1. * The
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle ...
has trilinears \cos A + 2 \cos B \cos C : \cos B + 2 \cos C \cos A : \cos C + 2 \cos A \cos B, corresponding to the parameter value t=2. * The
de Longchamps point In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let th ...
has trilinears \cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos B, corresponding to the parameter value t=-1.


Slope

In a
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, denote the slopes of the sides of a triangle as m_1, m_2, and m_3, and denote the slope of its Euler line as m_E. Then these slopes are related according toWladimir G. Boskoff, Laurent¸iu Homentcovschi, and Bogdan D. Suceava, "Gossard's Perspector and Projective Consequences", ''Forum Geometricorum'', Volume 13 (2013), 169–184

/ref> :m_1m_2 + m_1m_3 + m_1m_E + m_2m_3 + m_2m_E + m_3m_E :: + 3m_1m_2m_3m_E + 3 = 0. Thus the slope of the Euler line (if finite) is expressible in terms of the slopes of the sides as :m_E=-\frac. Moreover, the Euler line is parallel to an acute triangle's side ''BC'' if and only if \tan B \tan C = 3.


Relation to inscribed equilateral triangles

The locus of the centroids of
equilateral triangle 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 ...
s inscribed in a given triangle is formed by two lines perpendicular to the given triangle's Euler line.Francisco Javier Garc ́ıa Capita ́n, "Locus of Centroids of Similar Inscribed Triangles", ''
Forum Geometricorum ''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' was a peer-reviewed open-access academic journal that specialized in mathematical research papers on Euclidean geometry. Founded in 2001, it was published by Florida Atlantic Unive ...
'' 16, 2016, 257–267 .http://forumgeom.fau.edu/FG2016volume16/FG201631.pdf


In special triangles


Right triangle

In a
right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...
, the Euler line coincides with the
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
to the
hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...
—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its
altitudes Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometry, geographical s ...
, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.


Isosceles triangle

The Euler line of an
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
coincides with the
axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
. In an isosceles triangle the
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 ...
falls on the Euler line.


Automedian triangle

The Euler line of an automedian triangle (one whose
medians The Medes were an Iron Age Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, they occupied the mountainous region of northwestern Iran and ...
are in the same proportions, though in the opposite order, as the sides) is perpendicular to one of the medians..


Systems of triangles with concurrent Euler lines

Consider a triangle ''ABC'' with Fermat–Torricelli points ''F''1 and ''F''2. The Euler lines of the 10 triangles with vertices chosen from ''A, B, C, F''1 and ''F''2 are concurrent at the centroid of triangle ''ABC''. The Euler lines of the four triangles formed by an
orthocentric system In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through ...
(a set of four points such that each is the
orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...
of the triangle with vertices at the other three points) are concurrent at the
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle ...
common to all of the triangles.


Generalizations


Quadrilateral

In a convex quadrilateral, the quasiorthocenter ''H'', the "area centroid" ''G'', and the quasicircumcenter ''O'' are
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
in this order on the Euler line, and ''HG'' = 2''GO''.


Tetrahedron

A
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
is a
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
object bounded by four triangular faces. Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its Monge point; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter. These points define the "Euler line" of a tetrahedron analogous to that of a triangle. The centroid is the midpoint between its Monge point and circumcenter along this line. The center of the twelve-point sphere also lies on the Euler line.


Simplicial polytope

A
simplicial polytope In geometry, a simplicial polytope is a polytope whose facet_(mathematics), facets are all Simplex, simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only Triangle, triangular facessimplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
(plural of simplex). For example, every polygon is a simplicial polytope. The Euler line associated to such a polytope is the line determined by its centroid and circumcenter of mass. This definition of an Euler line generalizes the ones above.. Suppose that P is a polygon. The Euler line E is sensitive to the symmetries of P in the following ways: # If P has a line of reflection symmetry L, then E is either L or a point on L. # If P has a center of rotational symmetry C, then E=C.


Related constructions

A triangle's Kiepert parabola is the unique parabola that is tangent to the sides (two of them extended) of the triangle and has the Euler line as its directrix.Scimemi, Benedetto, "Simple Relations Regarding the Steiner Inellipse of a Triangle", ''Forum Geometricorum'' 10, 2010: 55–77.
/ref>


References


External links



* ttp://demonstrations.wolfram.com/EulerLine/ "Euler Line"an
"Non-Euclidean Triangle Continuum"
at the
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...

Nine-point conic and Euler line generalization
an

a

* Bogomolny, Alexander,
Altitudes and the Euler Line
and
Euler Line and 9-Point Circle
, ''
Cut-the-Knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
'' * * Archived a
Ghostarchive
and th
Wayback Machine
* {{mathworld , title = Euler Line , urlname = EulerLine Straight lines defined for a triangle