Euler Line

The line on which the Orthocenter $H$, Centroid $M$, Circumcenter $O$, de Longchamps Point $L$, Nine-Point Center $F$, and the Tangential Triangle Circumcircle $O_T$ of a Triangle lie. The Incenter lies on the Euler line only if the Triangle is an Isosceles Triangle. The Euler line consists of all points with Trilinear Coordinates $\alpha:\beta:\gamma$ which satisfy

$$\left\vert\matrix{\alpha & \beta & \gamma\cr \cos A & \cos B... ...r \cos B\cos C & \cos C\cos A & \cos A\cos B\cr}\right\vert=0,$$ (1)

which simplifies to

$\alpha\cos A(\cos^2 B-\cos^2 C)+\beta\cos B(\cos^2 C-\cos^2 A)$
$+\gamma\cos C(\cos^2 A-\cos^2 B)=0.\quad$	(2)

This can also be written

$$\alpha\sin(2A)\sin(B-C)+\beta\sin(2B)\sin(C-A)+\gamma\sin(2C)\sin(A-B)=0.$$ (3)

The Euler line may also be given parametrically by

$$P(\lambda)=O+\lambda H$$ (4)

(Oldknow 1996).

$\lambda$	Center
$-2$	point at infinity
$-1$	de Longchamps Point $L$
0	Circumcenter $O$
1	Centroid $G$
2	Nine-Point Center $F$
$\infty$	Orthocenter $H$

The Orthocenter is twice as far from the Centroid as is the Circumcenter. The Circumcenter $O$, Nine-Point Center $F$, Centroid $G$, and Orthocenter $H$ form a Harmonic Range.

The Euler line intersects the Soddy Line in the de Longchamps Point, and the Gergonne Line in the Evans Point. The Isotomic Conjugate of the Euler line is called Jerabek's Hyperbola (Casey 1893, Vandeghen 1965).

See also Centroid (Triangle), Circumcenter, Evans Point, Gergonne Line, Jerabek's Hyperbola, de Longchamps Point, Nine-Point Center, Orthocenter, Soddy Line, Tangential Triangle

References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 18-20, 1967.

Dörrie, H. ``Euler's Straight Line.'' §27 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 141-142, 1965.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 117-119, 1990.

Oldknow, A. ``The Euler-Gergonne-Soddy Triangle of a Triangle.'' Amer. Math. Monthly 103, 319-329, 1996.

Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.