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K458

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X(2), X(5), X(6), X(13), X(14), X(15), X(16), X(3070), X(3071)

For any point M on the Brocard axis, the isogonal conjugate M* of M lies on the Kiepert hyperbola and the line MM* envelopes the Brocard-Kiepert quartic Q073.

This line MM* meets the Kiepert hyperbola again at N*. The tangents at M* and N* to the Kiepert hyperbola meet at P and, when M traverses the Brocard axis, the locus of P is the Brocard-Kiepert cubic K458.

K458 is an unicursal cubic with an isolated point X(5) i.e. an acnodal cubic. It has three real inflexion points which are the Lemoine point K and the isodynamic points X(15), X(16). The tangents at X(15), X(16) meet at X on the Euler line such that OX = 3/4 OX(5) or GX = 1/4 GX(5).

The isogonal conjugates G, X(13), X(14) of these inflexion points are the contacts of the cubic with the Kiepert hyperbola. In other words, K458 is tritangent to the Kiepert hyperbola at these points.

K458 meets the Evans conic at six identified points namely X(13), X(14), X(15), X(16), X(3070), X(3071). The tangents at X(3070), X(3071) – two points on the line HK – pass through G.