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X(3), X(4), X(32), X(56), X(1147), E(403), E(908), E(910), E(912), E(914), E(935), E(936), E(992), E(994), E(996), E(998), E(1000), E(1002)

vertices of the circumnormal triangle

infinite points of the Orthocubic K006

projections of O on the altitudes

A', B', C' : midpoints of ABC

points Ua, Ub, Uc quoted in the Neuberg cubic page. See also table 16 and table 18.

In 1891, Lemoine published a note in which he very briefly studied a cubic curve defined as follows. Let M be a point in the plane of triangle ABC. Denote by Ma the intersection of the line MA with the perpendicular bisector of BC and define Mb, Mc similarly. The locus of M such that the three points Ma, Mb, Mc are collinear is the Lemoine cubic K009.

More informations and generalizations are found in the FG paper "The Lemoine Cubic and Its Generalizations" where the Lemoine cubic is denoted by K(O).

K009 is the Psi-transform of the circumcircle and the Phi-transform of the line at infinity. See CL037.

The isogonal transform of K009 is K028, the Musselman (third) cubic.

K009 is closely related to the quartic Q023. It is a member of the class CL033 (Deléham cubics).

For any point Q on the line X(2)X(6), the trilinear polar of Q meets the perpendicular bisectors at Qa, Qb, Qc. ABC and QaQbQc are perspective and the perspector is a point on K009. This gives a simple way to find a lot of reasonably simple points on the curve.

K009 is also psK(X184, X2, X3) in Pseudo-Pivotal Cubics and Poristic Triangles.