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X(1), X(3), X(4), X(56), X(3417), X(10571), X(11101), X(15446), X(17104)

X(58737) → X(58745)

imaginary foci of the MacBeath inconic

infinite points of pK(X6,X355)

vertices A', B', C' of the 2nd circum-perp triangle = circum-cevian triangle of X(1)

Geometric properties :

K1340 is a MacBeath cubic as in Table 80.

Following the notations of CL055, P = X(355) is the reflection of P' = X(1) in X(5) and P* = X(3417) is the isogonal conjugate of X(355).

Let La be the parallel in P' to the line AP. Then, La meets BC at U and AP* at U' which are two points on K1340.

V, W, V', W' are defined cyclically and these six points lie on a same (green) conic passing through X(11).

The triangles UVW and U'V'W' are perspective at X(1) and UVW and A'B'C' are perspective at X(58742), on the lines {1, 3417}, {3, 1854}, {4, 36}, {56, 11334}, {355, 2222}, etc.

Note that X(58737) = g X(13746) and X(15446) lie on the Jerabek and Feuerbach hyperbolas respectively.

K1340 and pK(X6,X355) meet at three points at infinity and six points on the circum-conic with perspector X(654), namely A, B, C, X(3417) and X(1) counted twice. X(1) is a point of inflexion with inflexional tangent passing through X(5), X(11).

See the related cubics K850, K1344 and K1345, the isogonal transform of K1340.