Hi everyone on Math Stackexchange,
I'm recently interested in the geometry of a triangle, and my studies now seems to require some knowledge on cubic curves related to a triangle, in particular the Darboux cubic. I could not find any useful reference of its properties' proofs (e.g. the de Longchamps points is its pivot, which means any point on the curve is collinear with its isogonal conjugate with respect to the reference triangle and the de Longchamps point of the reference triangle). Can any one please tell me where I can find discussions on the Darboux cubic related to a triangle please? Thank you.
Wilson Zhao
The fact that the de Longchamps point $L$ is the pivot of the Darboux cubic just follows from the fact that $L$ has trilinear coordinates: $$ L:[\cos A-\cos B\cos C;\cos B-\cos A\cos C;\cos C-\cos A\cos B]$$ and a point with trilinear coordinates $[\alpha;\beta;\gamma]$ lies on the Darboux cubic iff: $$ \sum_{cyc}(\cos A-\cos B\cos C)\alpha(\beta^2-\gamma^2)=0.$$ Good references for triangle cubics are Forum Geometricorum and the Hyacinthos mailing list.