Editing Euler line (section)

===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>