The de Longchamps ellipse of a triangle Δ A B C 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(α + β - γ)(α - β + γ) + b(α - β + γ)(α + β - γ) + c(α - β + γ)(α + β - γ) = 0, which can also be written (a - b - c) α^2 + (-a + b - c) β^2 + (-a - b + c) γ^2 + 2(a βγ + b αγ + c αβ) = 0.
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