Editing Euler line (section)

==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]]; 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>

An equation for the Euler line in [[barycentric coordinates (mathematics)|barycentric coordinates]] <math>\alpha :\beta :\gamma</math> is<ref>Scott, J.A., "Some examples of the use of areal coordinates in triangle geometry", ''Mathematical Gazette'' 83, November 1999, 472-477.</ref>

:<math>(\tan C -\tan B)\alpha +(\tan A -\tan C)\beta + (\tan B -\tan A)\gamma =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 [[circumcenter]] has trilinears <math>\cos A:\cos B:\cos C,</math> corresponding to the parameter value <math>t=0.</math>
* 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>