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de Longchamps Point


deLongchampsPoint

The de Longchamps point L is the reflection of the orthocenter H about the circumcenter O of a triangle. It has triangle center function

 alpha=cosA-cosBcosC,
(1)

and is Kimberling center X_(20) (Kimberling 1998, p. 70).

As a result of its definition, the de Longchamps point is collinear with the orthocenter H and circumcenter O of a triangle.

Distances to some other named triangle centers include

LG=4/3OH
(2)
LGe=(2(a^3-ba^2-ca^2-b^2a-c^2a-2bca+b^3+c^3-bc^2-b^2c)IL)/((a+b+c)(a^2-2ba-2ca+b^2+c^2-2bc))
(3)
LH=2OH
(4)
LI=1/(2r)(sqrt(a^4-ba^3-ca^3+bca^2-b^3a-c^3a+bc^2a+b^2ca+b^4+c^4-bc^3-b^3c))
(5)
LN=3/2OH
(6)
LO=OH,
(7)

where G is the triangle centroid, O is the circumcenter, H is the orthocenter, Ge is the Gergonne point, H is the orthocenter, I is the incenter, N is the nine-point center, and r is the incenter.

deLongchampsOrthocenter

The de Longchamps point is also the orthocenter of the anticomplementary triangle.

deLongchampsLines

The Soddy line intersects the Euler line in the de Longchamps point (Oldknow 1996).

The de Longchamps point and Kimberling center X_(650) (intersection L_G union L_0 of the Gergonne line and orthic axis) form a diameter of the GEOS circle.


See also

Anticomplementary Triangle, Circumcenter, Circumcircle, Euler-Gergonne-Soddy Triangle, Euler Line, GEOS Circle, Orthic Axis, Orthocenter, Soddy Line

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References

Altshiller-Court, N. "On the de Longchamps Circle of the Triangle." Amer. Math. Monthly 33, 368-375, 1926.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(20)=De Longchamps Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X20.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.Vandeghen, A. "Soddy's Circles and the de Longchamps Point of a Triangle." Amer. Math. Monthly 71, 176-179, 1964.

Referenced on Wolfram|Alpha

de Longchamps Point

Cite this as:

Weisstein, Eric W. "de Longchamps Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/deLongchampsPoint.html

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