Longchamps point

Longchamps point L

The Longchamps point (point of De Longchamps), named after the French mathematician Gohierre de Longchamps (1842–1906), is one of the excellent points of a triangle . It is defined as the mirror point (L) of the vertical intersection (H) at the circumcenter (U).

properties

The Longchamps point is on Euler's straight line .
The Longchamps point lies on a straight line with the Gergonne point and the center of the circle .

Longchamps point ( ) ${\ displaystyle X_ {20}}$
Trilinear coordinates	${\ displaystyle (\ cos \ alpha - \ cos \ beta \ cos \ gamma) \,: \, (\ cos \ beta - \ cos \ gamma \ cos \ alpha) \,: \, (\ cos \ gamma - \ cos \ alpha \ cos \ beta)}$
Barycentric coordinates	${\ displaystyle (\ tan \ beta + \ tan \ gamma - \ tan \ alpha) \,: \, (\ tan \ gamma + \ tan \ alpha - \ tan \ beta) \,: \, (\ tan \ alpha + \ tan \ beta - \ tan \ gamma)}$ ${\ displaystyle = f (a, b, c): f (b, c, a): f (c, a, b) {\ mbox {with}} f (a, b, c) = - 3a ^ { 4} + 2a ^ {2} (b ^ {2} + c ^ {2}) + (b ^ {2} -c ^ {2}) ^ {2}}$

A. Vandeghen: Soddy's Circles and the De Longchamps Point of a Triangle . The American Mathematical Monthly, Vol. 71, No. 2 (Feb., 1964), pp. 176-179 ( JSTOR 2311750 )