The line on which the orthocenter , triangle centroid , circumcenter , de Longchamps point , nine-point center , and a number of other important triangle centers lie.
The Euler line is perpendicular to the de Longchamps line and orthic axis.
Kimberling centers lying on the line include (triangle centroid ), 3 (circumcenter ), 4 (orthocenter ), 5 (nine-point center ), 20 (de Longchamps point ), 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 which satisfy
(1)
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which simplifies to
(2)
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This can also be written
(3)
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Another nice trilinear equation for the Euler line is given by
(4)
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where is aConway triangle notation. It is central line .
The Euler line satisfies the remarkable property of being its own complement, and therefore also its own anticomplement.
The circumcenter , nine-point center , triangle centroid , and orthocenter form a harmonic range with
(5)
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(6)
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(7)
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(8)
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(Honsberger 1995, p. 7; Oldknow 1996). Here, is the circumcenter-orthocenter distance, given by
(9)
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(10)
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(11)
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where is the circumradius and 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 for , 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 (, ) (P. Moses, pers. comm., Feb. 4, 2005).
For a point lying on the Euler line with trilinear coordinates
(12)
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the sum of squared distances from the vertices of the reference triangle equals
(13)
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(14)
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where is the circumradius, is the circumcenter, and 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 refers to the line passing through Kimberling centers and .
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).