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)
|
which simplifies to
 |
(2)
|
This can also be written
 |
(3)
|
Another nice trilinear equation for the Euler line is given by
 |
(4)
|
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)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
(Honsberger 1995, p. 7; Oldknow 1996). Here, 
is the circumcenter-orthocenter
distance, given by
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
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)
|
the sum of squared distances from the vertices of the reference triangle equals
 |  |  |
(13)
|
 |  |  |
(14)
|
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).
