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[[Image:Triangle.EulerLine.svg|thumb|right|300px|Euler's line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).]]

In [[geometry]], the '''Euler line''', named after [[Leonhard Euler]] (US {{IPAc-en|ˈ|ɔɪ|l|ər}}, {{respell|OY|lər}} or UK {{IPAc-en|ˈ|ɔɪ|l|ə}}, {{respell|OY|lə}}), is a [[line (mathematics)|line]] determined from any [[triangle]] that is not [[equilateral triangle|equilateral]]. It passes through several important points determined from the triangle, including the [[orthocenter]], the [[circumcenter]], the [[centroid]], the [[Exeter point]] and the center of the [[nine-point circle]] of the triangle.<ref name="k">{{cite journal
 | author = Kimberling, Clark
 | title = Triangle centers and central triangles
 | journal = Congressus Numerantium
 | volume = 129
 | year = 1998
 | pages = i–xxv, 1–295}}</ref>

==Triangle centers==
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are [[Line (geometry)|collinear]].<ref>{{cite journal
 | author = Euler, Leonhard
 | authorlink = Leonhard Euler
 | title = Solutio facilis problematum quorundam geometricorum difficillimorum
 | journal = Novi Commentarii academiae scientarum imperialis Petropolitanae
 | volume = 11
 | year = 1767
 | pages = 103–123
 | url = http://math.dartmouth.edu/~euler/pages/E325.html
 | id = <!--Enestrom number-->E325}} Reprinted in ''Opera Omnia'', ser. I, vol. XXVI, pp.&nbsp;139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, {{MR|0061061}}.
</ref> 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]], and the [[Exeter point]].<ref name="k"/> However, the [[incenter]] generally does not lie on the Euler line;<ref>{{cite book | url=http://books.google.com/books?id=lR0SDnl2bPwC&pg=PA4 | title=Geometry Turned On: Dynamic Software in Learning, Teaching, and Research | publisher=The Mathematical Association of America | author=Schattschneider, Doris; King, James | year=1997 | pages=3-4 | isbn=978-0883850992}}</ref> it is on the Euler line only for [[isosceles triangle]]s, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.

==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]]; then an equation for the Euler line is

:<math>\sin 2A \sin(B - C)x + \sin 2B \sin(C - A)y + \sin 2C \sin(A - B)z = 0.\,</math>

==Parametric representation==

Another way to represent the Euler line is in terms of a parameter ''t''.  Starting with the circumcenter (with trilinear coordinates <math>\cos A : \cos B : \cos C</math>) and the orthocenter (with trilinears <math>\sec A : \sec B : \sec C = \cos B \cos C : \cos C \cos A : \cos A \cos B)</math>, every point on the Euler line, except the orthocenter, is given by the trilinear coordinates
:<math>\cos A + t \cos B \cos C : \cos B + t \cos C \cos A : \cos C + t \cos A \cos B\,</math>
formed as a [[linear combination]] of the trilinears of these two points, for some ''t''.

For example:
* The [[centroid]] has trilinears <math>\cos A + \cos B \cos C : \cos B + \cos C \cos A : \cos C + \cos A \cos B</math>, corresponding to the parameter value <math>t=1</math>.
* The [[nine-point center]] has trilinears <math>\cos A + 2 \cos B \cos C : \cos B + 2 \cos C \cos A : \cos C + 2 \cos A \cos B</math>, corresponding to the parameter value <math>t=2</math>.
* The [[De Longchamps point]] has trilinears <math>\cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos B</math>, corresponding to the parameter value <math>t=-1</math>.

==Slope==

In a [[Cartesian coordinate system]], denote the slopes of the sides of a triangle as <math>m_1,</math> <math>m_2,</math> and <math>m_3,</math> and denote the slope of its Euler line as <math>m_E</math>.  Then these slopes are related according to<ref name=BHS>Wladimir G. Boskoff, Laurent¸iu Homentcovschi, and Bogdan D. Suceava, "Gossard’s Perspector and Projective Consequences", ''Forum Geometricorum'', Volume 13 (2013), 169–184. [http://forumgeom.fau.edu/FG2013volume13/FG201318.pdf]</ref>{{rp|Lemma 1}}

:<math>m_1m_2 + m_1m_3 + m_1m_E + m_2m_3 + m_2m_E + m_3m_E</math>
::<math> + 3m_1m_2m_3m_E + 3 = 0.</math>

Thus the slope of the Euler line (if finite) is expressible in terms of the slopes of the sides as

:<math>m_E=-\frac{m_1m_2 + m_1m_3 + m_2m_3 + 3}{m_1 + m_2 + m_3 + 3m_1m_2m_3}.</math>

Moreover, the Euler line is parallel to an acute triangle's side ''BC'' if and only if<ref name=BHS/>{{rp|p.173}} <math>\tan B \tan C = 3.</math>

==Lengths of segments==

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:<ref name="ac">Altshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. Barnes & Noble 1952).</ref>{{rp|p.102}}

:<math>GH=2GO;</math>
:<math>OH=3GO.</math>

The center of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter.<ref name="k"/> 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''<sup>2</sup> by an amount equal to one-ninth the sum of the squares of the side lengths ''a'', ''b'', and ''c'':<ref name="ac"/>{{rp|p.71}}

:<math>GO^2=R^2-\tfrac{1}{9}(a^2+b^2+c^2).</math>

==Right triangle==

In a [[right triangle]], the Euler line contains the [[median (triangle)|median]] on the [[hypotenuse]]&mdash;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 [[Altitude (triangle)|altitudes]], falls on the right-angled vertex while its circumcenter, the intersection of its [[Bisection#Perpendicular bisectors|perpendicular bisectors]] of sides, falls on the midpoint of the hypotenuse.

==See also==
*[[Gossard perspector]]
*[[Central line (geometry)|Central line]]

== References ==
{{reflist}}

== External links ==
* [http://www.cut-the-knot.org/triangle/altEuler.shtml Altitudes and the Euler Line] and [http://www.cut-the-knot.org/triangle/EulerLine.shtml Euler Line and 9-Point Circle] at [[cut-the-knot]]
* [http://faculty.evansville.edu/ck6/tcenters/class/eulerline.html Triangle centers on the Euler line], by Clark Kimberling.
* [http://www.mathopenref.com/eulerline.html An interactive applet showing several triangle centers that lies on the Euler line].
* {{mathworld | title = Euler Line | urlname = EulerLine}}
* [http://demonstrations.wolfram.com/EulerLine/ "Euler Line"] and [http://demonstrations.wolfram.com/NonEuclideanTriangleContinuum/ "Non-Euclidean Triangle Continuum"] at the [[Wolfram Demonstrations Project]]
* [http://dynamicmathematicslearning.com/ninepointconic.html Nine-point conic and Euler line generalization] and  [http://dynamicmathematicslearning.com/furtherEuler.html A further Euler line generalization]  at Dynamic Geometry Sketches

[[Category:Triangle geometry]]