The de Longchamps point is the reflection of the orthocenter about the circumcenter of a triangle. It has triangle center function
(1)
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and is Kimberling center (Kimberling 1998, p. 70).
As a result of its definition, the de Longchamps point is collinear with the orthocenter and circumcenter of a triangle.
Distances to some other named triangle centers include
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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where is the triangle centroid, is the circumcenter, is the orthocenter, is the Gergonne point, is the orthocenter, is the incenter, is the nine-point center, and is the incenter.
The de Longchamps point is also the orthocenter of the anticomplementary triangle.
The Soddy line intersects the Euler line in the de Longchamps point (Oldknow 1996).
The de Longchamps point and Kimberling center (intersection of the Gergonne line and orthic axis) form a diameter of the GEOS circle.