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Euler Line


EulerLine

The line on which the orthocenter H, triangle centroid G, circumcenter O, de Longchamps point L, nine-point center N, and a number of other important triangle centers lie.

The Euler line is perpendicular to the de Longchamps line and orthic axis.

Kimberling centers X_i lying on the line include i=2 (triangle centroid G), 3 (circumcenter O), 4 (orthocenter H), 5 (nine-point center N), 20 (de Longchamps point L), 21 (Schiffler point), 22 (Exeter point), 23 (far-out point), 24, 25, 26, 27, 28, 29, 30, (Euler infinity point), 140, 186, 199, 235, 237, 297, 376, 377, 378, 379, 381, 382, 383, 384, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 546, 547, 548, 549, 550, 631, 632, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 964, 1003, 1004, 1005, 1006, 1008, 1009, 1010, 1011, 1012, 1013, 1080, 1113, 1114, 1312, 1313, 1314, 1315, 1316, 1325, 1344, 1345, 1346, 1347, 1368, 1370, 1375, 1513, 1529, 1532, 1536, 1551, 1556, 1557, 1559, 1563, 1564, 1567, 1583, 1584, 1585, 1586, 1589, 1590, 1591, 1592, 1593, 1594, 1595, 1596, 1597, 1598, 1599, 1600, 1628, 1650, 1651, 1656, 1657, 1658, 1816, 1817, 1883, 1884, 1885, 1889, 1894, 1904, 1906, 1907, 1981, 1982, 1984, 1985, 1995, 2041, 2042, 2043, 2044, 2045, 2046, 2047, 2048, 2049, 2050, 2060, 2070, 2071, 2072, 2073, 2074, 2075, 2409, 2450, 2454, 2455, 2475, 2476, 2478, 2479, 2480, 2552, 2553, 2554, 2555, 2566, 2567, 2570, 2571, 2675, 2676, 2915, and 2937.

The Euler line consists of all points with trilinear coordinates alpha:beta:gamma which satisfy

 |alpha beta gamma; cosA cosB cosC; cosBcosC cosCcosA cosAcosB|=0,
(1)

which simplifies to

 alphacosA(cos^2B-cos^2C)+betacosB(cos^2C-cos^2A)+gammacosC(cos^2A-cos^2B)=0.
(2)

This can also be written

 alphasin(2A)sin(B-C)+betasin(2B)sin(C-A)+gammasin(2C)sin(A-B)=0.
(3)

Another nice trilinear equation for the Euler line is given by

 a(b^2-c^2)S_Ax+b(c^2-a^2)S_By+c(a^2-b^2)S_Cz=0,
(4)

where S_x is aConway triangle notation. It is central line L_(647).

The Euler line satisfies the remarkable property of being its own complement, and therefore also its own anticomplement.

EulerLineHarmonicRange

The circumcenter O, nine-point center N, triangle centroid G, and orthocenter H form a harmonic range with

GO=1/2HG
(5)
OG=1/3OH
(6)
ON=1/2OH
(7)
NG=1/6HO
(8)

(Honsberger 1995, p. 7; Oldknow 1996). Here, OH is the circumcenter-orthocenter distance, given by

OH=(sqrt(a^6-b^2a^4-c^2a^4-b^4a^2-c^4a^2+3b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2))/(4Delta)
(9)
=sqrt(9R^2-(a^2+b^2+c^2))
(10)
=sqrt(9R^2-2S_omega),
(11)

where R is the circumradius and S_omega is Conway triangle notation.

The Euler line intersects the Soddy line in the de Longchamps point, and the Gergonne line in the Evans point.

The isogonal conjugate of the Euler line is the Jerabek hyperbola (Casey 1893, Vandeghen 1965).

The isotomic conjugate of the Euler line is a circumhyperbola passing through Kimberling centers X_i for i=2, 69, 95, 253, 264, 287, 305, 306, 307, 328, 1441, 1494, 1799, 1972, 2373, and 2419. This circumhyperbola is also the isogonal conjugate of the line (X_6, X_(25)) (P. Moses, pers. comm., Feb. 4, 2005).

For a point P lying on the Euler line with trilinear coordinates

 aS_A+(kS_BS_C)/a:bS_B+(kS_CS_A)/b:cS_C+(kS_AS_B)/c,
(12)

the sum of squared distances from the vertices of the reference triangle equals

AP^2+BP^2+CP^2=3R^2+((k-4)kOH^2)/((2+k)^2)
(13)
=3R^2+((k-4)OP^2)/k,
(14)

where R is the circumradius, O is the circumcenter, and H is the orthocenter of the reference triangle (P. Moses, pers. comm., Feb. 23, 2005).

The following table summarizes the Euler lines of a number of named triangles (P. Moses, pers. comm.), where L_(i,j) refers to the line passing through Kimberling centers i and j.

The angle between the orthic axis and Gergonne line is equal to that between the Euler line and the Soddy line (F. Jackson, pers. comm., Nov. 2, 2005).


See also

Central Line, Circumcenter, Euler-Gergonne-Soddy Triangle, Evans Point, Gergonne Line, Jerabek Hyperbola, de Longchamps Point, Nine-Point Center, Orthocenter, Soddy Line, Tangential Triangle, Triangle Centroid

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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. "The Medial Triangle and Euler Line." §1.7 in 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.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 28, 1928.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 7, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 69, 1991.

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Euler Line

Cite this as:

Weisstein, Eric W. "Euler Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerLine.html

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