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A MacBeath cubic is a circum-cubic which passes through the four foci of the MacBeath inconic, namely O, H and two imaginary isogonal points on the perpendicular bisector of OH and on the circum-conic passing through X(54), X(110) which is its isogonal transform. Every MacBeath cubic MB(P) is a spK(P, X5) for some point P. See CL055 for spK cubics. All these cubics are in a same net which contains lots of remarkable cubics. Some of these cubics are already mentioned in Table 54, see line Q = X(5) in the table and note 3. These are the cubics obtained when P lies on the Euler line. This page presents a more in-depth study. Recall that MB(P) passes through the isogonal conjugate P* of P and the reflection P' of P in X(5). MB(P), pK(X6, P) share the same points at infinity and MB(P), pK(X6, P') share the same points on (O). MB(P) and MB(P') are isogonal transforms of one and another. They coincide if and only if P = X(5) giving the Napoleon cubic K005, or P lies on the line at infinity, giving the focal isogonal nKs mentioned below. More generally, for every P on K005, MB(P) passes through P and obviously also through P*, also on K005, and P'.
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MB(P) is an equilateral cubic if and only if P = O, giving the McCay stelloid K028 = psK(X4, X264, X3).
MB(P) is a circular cubic if and only if P lies on the line at infinity. In this case, it is an isogonal focal nK with singular focus on the circumcircle. Its root lies on the trilinear polar of X(264), passing through X(297), X(525), X(850), X(2501), X(2592), X(2593), etc. Every cubic is the locus of foci of inscribed conics centered on a line passing through X(5). K164 is the only nK0 of this type.
MB(P) is a K0 (no term in xyz) if and only if P lies on the (blue) line passing through X(5), X(6) and many other centers.
MB(P) is a psK if and only if P lies on K044 = pK(X216, X4), the Darboux cubic of the orthic triangle. In this case, the pseudo-pole Ω lies on K176 = pK(X32, X4) and the pseudo-pivot Q lies on K674 = pK(X324, X264). It follows that MB(P) is a pK if and only if P is X(5), X(68), X(155) corresponding to the Napoleon cubic K005, K1318 = pK(X571, X4), K1337 = pK(X2165, X847) respectively (see yellow cells in the table below). |
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MB(P) is a nK if and only if P lies on the line at infinity as above, or on the (green) circum-conic with center X(5), perspector X(216), called the Johnson circum-conic. In this case, its pole Ω lies on the nodal cubic cK(#X6, X4) and its root lies on the circum-conic with perspector X(5). When P is A, B, C, X(110), X(265), the nK decomposes into a line and a conic. When P is a reflection A', B', C' of A, B, C in X(5), the nK is a nodal cubic with node A, B, C respectively. The isogonal transform of spK(A', X5) is a rectangular hyperbola with center X(5), passing through A, A', the four foci of the MacBeath inconic, the infinite points of the A-bisectors.
MB(P) is a K+ if and only if P lies on a (orange) cubic passing through X(3), X(54), X(382), X(15801), the infinite points of K005, the points on (O) of K003 (vertices N1, N2, N3 of the CircumNormal triangle) and on nK0(X6, X37367). These latter points are the antipodes M1, M2, M3 of the points of pK(X6, X1657) on (O), where X(1657) is the reflection of X(3) in X(20). This cubic meets K005 again at six finite points on the rectangular hyperbola passing through X(3), X(20), X(54), X(155), X(2574), X(2575).
MB(P) is a nodal cubic if and only if P lies on a (dashed brown) very complicated bicircular curve of degree 12, passing through X(3), X(4), X(110), X(265), X(952), the infinite points of the lines passing through X(5) and the excenters, the infinite points of the circum-conic with perspector X(577). This curve is symmetric about X(5).
The table below shows all the listed cubics and several other remarkable examples. F is the singular focus of a focal cubic. ∞Knnn are the infinite points of Knnn. |
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Focal nKs with root R (on the trilinear polar of X264), singular focus F (on the circumcircle), computed by Peter Moses |
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