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(x + y + z)^3 = 27 (- x + y + z)(x - y + z)(x + y - z)

or

9 (x^3 + y^3 + z^3 + 3 x y z) = 2(x + y + z)^3

X(2)

infinite points of the sidelines of ABC

K700 is the locus of centers of conics inscribed in ABC and tangent to the Steiner (circum) ellipse.

It is also the locus of centers of conics inscribed in ABC having the same area as that of the Steiner inscribed ellipse.

Jean-Pierre Ehrmann observes that it is also the locus of P such that the product of distances from P to the sidelines of the medial triangle is constant and equal to the same product from G.

K700 is the complement of K656 and also the image of K015 under the homothety h(G, -3/4).

K700 is an acnodal cubic with node G = X(2). It has three inflexional asymptotes which are the sidelines of the medial triangle.

Generalization 1 :

Let P = p:q:r be a point not lying on the sidelines of ABC and C(P) the circum-conic with perspector P. The locus of centers of conics inscribed in ABC and tangent to C(P) is the cubic K(P) with equation :

[(- p + q + r) x + (p - q + r) y + (p + q - r) z]^3 = 27 p q r (- x + y + z) (x - y + z) (x + y - z)

which is the complement of K'(P) with equation :

(p x + q y + r z)^3 = 27 p q r x y z.

Hence, K(X2) = K700 and K'(X2) = K656. Also K'(X6) = K244.

K(P) is an acnodal cubic with node ctP, the center of the inconic I(P) with perspector P.

It has three inflexion points Fa, Fb, Fc lying on the sidelines of the medial triangle and on the trilinear polar L(P) of taP with equation :

(- p + q + r) x + (p - q + r) y + (p + q - r) z = 0.

These points are :

Fa = q-r:q:-r, Fb = p:p-r:-r and Fc = p:-q:p-q.

The inflexional tangents are the sidelines of the medial triangle.

The harmonic polars are the cevian lines of ctP in the medial triangle.

The Hessian H(P) of K(P) is also a nodal cubic with node ctP, obviously passing through Fa, Fb, Fc and tangent to the sidelines of the medial triangle at their meets with the cevian lines of ctP in this triangle and also with the cevian lines of P in ABC. These are the points :

Ga = q+r:q:r, Gb = p:p+ r:r and Gc = p:q:p+q.

Generalization 2 :

Let P = p:q:r be a finite point not lying on the sidelines of ABC and I(P) the in-ellipse with center P. The locus of centers of ellipses inscribed in ABC and having the same area as I(P) is the cubic K2(P) with equation :

(p + q + r)^3 (x^3 + y^3 + z^3 + 3 x y z) = (p^3 + q^3 + r^3 + 3 p q r) (x + y + z)^3.

P must lie in one of the four regions delimited by the sidelines of the medial triangle and containing G or one of the vertices of the antimedial triangle in order to have an ellipse.

K2(P) contains P and has 3 inflexional asymptotes which are the sidelines of the medial triangle.

K2(X2) is K700 corresponding to ellipses having the same area as the Steiner in-ellipse. It is the only cubic of this type which is a nodal cubic.

References :

Terquem : Nouvelles annales de mathématiques 1^re série, tome 4 (1845), p. 480-491, specially § XLIX.

Mention : Nouvelles annales de mathématiques 1^re série, tome 9 (1850), p. 5-9, see Lemme 4.

Lemoyne : Lieux géométriques, p.41, loci 262 and 315.