The Bickart points are the foci F_1 and F_2 of the Steiner circumellipse. They have trilinear coordinates α_1 :β_1 :γ_1 and α_2 :β_2 :γ_2, where α_i | = | (sqrt(2)Δ)/a ± sqrt(-a^2 + Z + b c cos(B - C)) β_i | = | (sqrt(2)Δ)/b ± sqrt(-b^2 + Z + a c cos(A - C)) γ_i | = | (sqrt(2)Δ)/c ± sqrt(-c^2 + Z + a b cos(A - B)) where Z = sqrt(a^4 + b^4 + c^4 - a^2 b^2 - b^2 c^2 - c^2 a^2). and for some appropriate choices of signs.
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