In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Euler line, named after
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
(), is a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
determined from any
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
that is not
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. It is a
central line of the triangle, and it passes through several important points determined from the triangle, including the
orthocenter, the
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, 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 surface of the figure. The same definition extends to any ...
, the
Exeter point
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a c ...
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 ea ...
of the triangle.
The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
and the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
.
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 points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
. This property is also true for another
triangle center, the
nine-point center, 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, the
Schiffler point, the
Exeter point
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a c ...
, 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 sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
The
tangential triangle of a reference triangle is tangent to the latter's
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
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
In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is ''extern ...
of the orthic and tangential triangles is also on the Euler line.[.]
A vector proof
Let be a triangle. A proof of the fact that the circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, 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 surface of the figure. The same definition extends to any ...
and the orthocenter are collinear relies on free vectors. We start by stating the prerequisites. First, satisfies the relation
:
This follows from the fact that the absolute barycentric coordinates of are . Further, the problem of Sylvester[Dörrie, 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
:
Now, using the vector addition, we deduce that
:
By adding these three relations, term by term, we obtain that
:
In conclusion, , and so the three points , and (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.
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).]
:
:
The segment ''GH'' is a diameter of the orthocentroidal circle
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a sub ...
.
The center ''N'' of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter:
:
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
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every pol ...
''R''2 by an amount equal to one-ninth the sum of the squares of the side lengths ''a'', ''b'', and ''c'':
:
In addition,
:
:
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 t ...
; then an equation for the Euler line is
:
An equation for the Euler line in barycentric coordinates is
:
Parametric representation
Another way to represent the Euler line is in terms of a parameter ''t''. Starting with the circumcenter (with trilinear coordinates ) and the orthocenter (with trilinears every point on the Euler line, except the orthocenter, is given by the trilinear coordinates
:
formed as a linear combination of the trilinears of these two points, for some ''t''.
For example:
* The circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
has trilinears corresponding to the parameter value
* 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 surface of the figure. The same definition extends to any ...
has trilinears corresponding to the parameter value
* The nine-point center has trilinears corresponding to the parameter value
* The de Longchamps point has trilinears corresponding to the parameter value
Slope
In a Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, denote the slopes of the sides of a triangle as and and denote the slope of its Euler line as . Then these slopes are related according to[Wladimir G. Boskoff, Laurent¸iu Homentcovschi, and Bogdan D. Suceava, "Gossard's Perspector and Projective Consequences", ''Forum Geometricorum'', Volume 13 (2013), 169–184]
/ref>
:
::
Thus the slope of the Euler line (if finite) is expressible in terms of the slopes of the sides as
:
Moreover, the Euler line is parallel to an acute triangle's side ''BC'' if and only if[
]
Relation to inscribed equilateral triangles
The locus of the centroids of equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
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'' is a peer-reviewed open-access academic journal
An academic journal or scholarly journal is a periodical publication in which scholarship relating to a particular academic d ...
'' 16, 2016, 257–267 .http://forumgeom.fau.edu/FG2016volume16/FG201631.pdf
In special triangles
Right triangle
In a right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
, the Euler line coincides with the median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
to the hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
—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, 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 sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
coincides with the axis of symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. . 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
In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different or ...
(one whose medians
The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient 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, the ...
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 of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...
(a set of four points such that each is the orthocenter of the triangle with vertices at the other three points) are concurrent at the nine-point center 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 points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
in this order on the Euler line, and ''HG'' = 2''GO''.
Tetrahedron
A tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
is a three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
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 facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz ...
is a polytope whose facets are all simplices (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
In geometry, the circumcenter of mass is a center associated with a polygon which shares many of the properties of the center of mass. More generally, the circumcenter of mass may be defined for simplicial polytopes and also in the spherical and hy ...
. This definition of an Euler line generalizes the ones above.[.]
Suppose that is a polygon. The Euler line is sensitive to the symmetries of in the following ways:
1. If has a line of reflection symmetry , then is either or a point on .
2. If has a center of rotational symmetry , then .
3. If all but one of the sides of have equal length, then is orthogonal to the last side.
Related constructions
A triangle's Kiepert parabola is the unique parabola that is tangent to the sides (two of them extended
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
) 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
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-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
''
*
* Archived a
Ghostarchive
and th
Wayback Machine
* {{mathworld , title = Euler Line , urlname = EulerLine
Straight lines defined for a triangle