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K729

x^2 [b^2 c^2 (y - z) + 2 (c^4 y - b^4 z)] = 0

X(2), X(6), X(1383)

fixed points of the involution f in page K024

vertices of the Grebe triangle

foci of the Steiner inellipse

infinite points of pK(X6, X599)

K729 is a member of the pencil of cubics generated by the Grebe cubic K102 and the union of the circumcircle and the line GK.

It is the only one that contains the four foci of the Steiner inellipse and also the four fixed points of the involution f as in page K024. Recall that these points lie on the Stammler hyperbola and that the diagonal triangle of these four points is the CircumTangential triangle.

K729 meets :

• the line at infinity at the same points as pK(X6, X599) or pK(X2, X7850).

• the circumcircle at the same points as K102 = pK(X6, X6). See also Table 57.

• the Steiner circum-ellipse at the same points as pK(X2, X7766).

• the sidelines of ABC at A' = 0 : 2a^2 + b^2 : 2a^2 + c^2, B' and C' likewise. A' lies on the parallel at X(6) to the line AX(599).

• the median AG at Ag = a^2 : 2(b^2 + c^2) : 2(b^2 + c^2) and the symmedian AK at Ak = -2a^2 : b^2 : c^2.

The isogonal transform of K729 is K287 = spK(X6, X2), a central circum-cubic with center G. See CL055. Hence K729 is spK(X599, X2).

The polar conic of X(2) in K729 is the rectangular hyperbola that also contains X(3), X(6), X(83), X(99), X(194), X(3413), X(3414). This is the Jerabek hyperbola of the Grebe triangle. In other words, the tangents at G1, G2, G3, X(6) pass through X(2). It follows that K729 also contains the vertices of the cevian triangle of X(6) in the Grebe triangle which is actually its orthic triangle. Hence, K729 is the pK with pivot X(6), isopivot X(2) in G1G2G3 which is pK(X4 x X54, X4) for the Grebe triangle.

Construction

Let (L) be a line passing through X(6), (L') its reflection about X(2) and (C') the isogonal transform of (L'). (L) and (C') meet at two points M, N on K729. Note that if (C) is the isogonal transform of (L) then (C) meets (L') at the isogonal conjugates of M, N on the central cubic spK(X6, X2).