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van Aubel Line


vanAubelLine

The van Aubel line is the line in the plane of a reference triangle that connects the orthocenter H and symmedian point K, and symmedian point of the orthic triangle. The collinearity of these three points is given as an exercise and ascribed to van Aubel by Casey (1888, Exercise 77, p. 241).

The van Aubel line is central line L_(520), and has trilinear equation

 alphacosA[sin(2B)-sin(2C)]+betacosB[sin(2C)-sin(2A)]+gammacosC[sin(2A)-sin(2B)],

which can also be written as

 a(b^2-c^2)S_A^2alpha+b(c^2-a^2)S_B^2beta+c(a^2-b^2)S_C^2gamma=0

(P. Moses, pers. comm., Mar. 24, 2005).

A complete list of Kimberling centers X_i through which it passes is given by i=4 (orthocenter H), 6 (symmedian point K), 53 (symmedian point of the orthic triangle), 217, 387, 393, 397, 398, 1172, 1181, 1199, 1249, 1498, 1503, 1514, 1515, 1540, 1547, 1548, 1549, 1587, 1588, 1834, 1865, 1901, 1990, 2207, 2211, 2442, and 2883.

It is perpendicular to lines (3,878), (30,511), (99,249), (110,935), (297,850), (323,401), and (441,647). It is parallel to lines (2,154), (3,66), (4,6), (5,182), (11,1428), (20,64), (22,161), (30,511), (51,428), (67,74), (98,230), (110,858), (125,468), (147,325), (184,427), (221,388), (242,1146), (265,1177), (287,297), (376,599), (381,597), (382,1351), (383,395), (394,1370), (396,1080), (546,575), (576,1353), (611,1478), (613,1479), and (946,1386).

The trilinear pole of the line is Kimberling center X_(107).


See also

Central Line

Portions of this entry contributed by Floor van Lamoen

Explore with Wolfram|Alpha

References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.

Referenced on Wolfram|Alpha

van Aubel Line

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "van Aubel Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/vanAubelLine.html

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