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K741

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X(30), X(325), X(523), X(1312), X(1313), X(1552)

U, V, W on the trilinear polar of X(525)

other points below

K741 is a conico-pivotal cubic with node X(523), an isolated point on the line at infinity. See a generalization in the page K406.

The pivotal conic is a rectangular hyperbola passing through X(5), X(10), X(141), X(2574), X(2575) hence homothetic to the Jerabek hyperbola.

The contact conic is the rectangular circum-hyperbola passing through X(4), X(5), X(53), X(311), X(327), X(1141), X(1263), X(1487), X(2165), X(2980).

K741 is the reciprocal polar transform of the Steiner deltoid H3 with respect to the parabola (P) which is the complement of the Kiepert parabola. (P) is a diagonal conic with respect to ABC and passes through X(523), X(656), X(661). It is inscribed in the medial triangle.

K741 has three inflexions F1, F2, F3 on the line passing through X(125), X(526), X(3134).

K741a

 

The pivotal conic and the contact conic meet at X(5) and three other points S1, S2, S3 lying on K741.

These are the vertices of the equilateral triangle which is the image of the CircumTangential triangle under the translation that maps X(3) onto X(5).

The sidelines of S1S2S3 are the Simson lines of the vertices of the CircumNormal triangle.

The three inflexions F1, F2, F3 lie on the sidelines of this triangle. They are the isoconjugates of S1, S2, S3 hence the tangents at S1, S2, S3 to the pivotal conic must pass through F1, F2, F3 respectively.

K741 has a real asymptote which is the parallel at X(110) to the Euler line. Its intersection with the cubic is X on the line X(523)X(3134), SEARCH = 4.24047260586700.