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∑ (b^2 - c^2)^2 (c^2 y - b^2 z) y z = 0

X(6), X(338), X(523), X(3613), X(7668), X(36198), X(36199), X(36200)

A'B'C' : cevian triangle of X(6)

A1B1C1 : anticevian triangle of X(523)

infinite points of the circum-hyperbola with perspector X(826)

other points below

Geometric properties :

Consider the Steiner inellipse (S) and the Orthic inconic (K), the inconics with centers X(2) and X(6) respectively.

Let M' = 𝛗(M) be the intersection of the polar lines of M in these two conics and obviously in any two conics of the pencil they generate.

If F is a fixed point, the locus of M such that F, M, M' are collinear is a cubic since 𝛗 is a quadratic involution.

This cubic is a circum-cubic if and only if F = X(523) and it is K1144 = pK(X115, X6).

K1144 must contain the singular points of 𝛗 which are the vertices of the anticevian triangle of X(523) and the fixed points of 𝛗 which are the four (not always real) common points of (S) and (K).

K1144 has three real asymptotes and one of them is the line X(6)X(523). The two other asymptotes are parallel to those of the circum-conic with perspector X(826) and meet at X = 𝛗(X9479). These asymptotes are also parallel to those of the 𝛗-transform (H) of the line at infinity, a hyperbola meeting K1144 again at X(6), A1, B1, C1 and also passing through X(2), X(427), X(570), X(1194), etc.

K1144 meets (S) again at two other points S1, S2 on the line X(2)X(6) and on K237 = pK(X115, X2).

K1144 meets (K) again at two other points K1, K2 on the line X(4)X(6) and on K238 = pK(X115, X4).

Note that {S1, S2} and {K1, K2} are two pairs of always real points which are swapped by the isoconjugation with pole X(115).

Since ABC and A1B1C1 are perspective at X(523), K1144 is also a pK with respect to A1B1C1 with pivot X(523) and isopivot X(6). Recall that X(115) is the barycentric square of X(523) hence A1, B1, C1 are the harmonic associates of X(523).

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Generalization

Let (P) and (Q) be the inconics with distinct centers P = p:q:r and Q = u:v:w. Define the quadratic involution 𝛗 as above.

The pole of the line PQ in the Steiner inellipse is F = (q - r) u - p (v- w) : : . Obviously, if one of the points P, Q is X(2), then F is the other.

The locus of M such that F, M, M' = 𝛗(M) are collinear is a pK with pole the barycentric square of F, passing through the common points of (P), (Q) and also the harmonic associates of F.

The pivot of this pK is Z = p u - (v - w) (q - r) : : . If Q = X(2) then Z = P.

If none of the points P, Q is X(2) then Z lies on the lines passing through :

• the barycentric products P x Q of P, Q and the infinite points of the trilinear polars of the isotomic conjugates of P, Q,

• the anticomplement of P x Q and the isotomic conjugate of the Ceva point of P, Q.

Z is the crosspoint of taP and taQ, where taX is the isotomic conjugate of the anticomplement of X.

See other properties at K554 when the inconics are given by their perspectors. K925 and K927 are two other related cubics.