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X(3), X(4), X(24), X(847), X(5889), X(16000), X(25044)

X(59274) → X(59281)

imaginary foci of the MacBeath inconic

infinite points of pK(X6,X18436)

vertices of the circum-orthic triangle HaHbHc

Geometric properties :

K1344 is a MacBeath cubic as in Table 80. Its isogonal transform is K1346.

K850 = spK(X382, X5) and K1340 = spK(X355, X5) are two cubics of Table 80 that meet (O) again at the vertices of the circum-cevian triangle of some point Q, namely X(3), X(1) respectively, which are points of inflexion on the cubics.

More generally, for any Q on the McCay cubic K003, one can always find a point P such that spK(P, X5) has the same property.

With Q = X(4), we find P = X(18436) and the cubic is K1344.

Here again, X(4) is a point of inflexion with inflexional tangent passing through X(265) and harmonic polar passing through X(186), X(523).

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Other points on K1344

• Z = X(59274) = barycentric product X(94) x X(30522), third point on the line {5,523} which passes through the imaginary foci of the MacBeath inconic.

• Y = X(59277), on the lines {3,16000}, {186,847}, {568,25044), perspector of triangles HaHbHc and A'B'C'. A' is the third point on the sideline BC, on the parallel at X(5889) to the line AX(18436).

• X(59281) = isogonal conjugate of the midpoint X(9927) of X(4), X(68).

• X(59276) = isogonal conjugate of X(58725).

• X(59278) = isogonal conjugate of X(18436).