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de Longchamps Ellipse


deLongchampsEllipse

The de Longchamps ellipse of a triangle DeltaABC is the conic circumscribed on the incentral triangle and the Cevian triangle of the isogonal mittenpunkt X_(57). (Since a conic is uniquely determined by five points, the conic is already specified with only five of these six points.)

The de Longchamps ellipse is centered at the incenter I of the reference triangle, and has trilinear equation

 a(alpha+beta-gamma)(alpha-beta+gamma)+b(alpha-beta+gamma)(alpha+beta-gamma)+c(alpha-beta+gamma)(alpha+beta-gamma)=0,

which can also be written

 (a-b-c)alpha^2+(-a+b-c)beta^2+(-a-b+c)gamma^2 
 +2(abetagamma+balphagamma+calphabeta)=0.

It passes through the points X_(244), X_(2170), and X_(2611) (Weisstein, Oct. 17 and Nov. 22, 2004).


See also

Cevian Triangle, de Longchamps Circle, Incentral Triangle, Isogonal Mittenpunkt

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References

Catalan, E. "Note sur l'ellipse de Longchamps." J. Math. Spéciales 4, 28-30, 1893.

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de Longchamps Ellipse

Cite this as:

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

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