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X(3), X(8), X(84), X(164), X(221)

infinite points of the altitudes

A', B', C' vertices of the cevian triangle of X(280)

Let (C) be the circum-conic with perspector P. For any point M on (C), let MaMbMc be the cevian triangle of M and let HM be its orthocenter. When M traverses (C) the locus of HM is a nodal cubic (K) passing through O = X(3) and having three real asymptotes parallel to the altitudes of ABC. The node N lies on the line through X(3) and the center of (C) i.e. the X(2)-Ceva conjugate of P.

N is the orthocenter of a triangle HaHbHc with vertices on the sidelines of ABC defined as follows. The perpendicular at the A-vertex Pa of the anticevian of P meets AB, AC at Mc, Mb. Ha is the intersection of the perpendiculars at Mb to PbMc and at Mc to PcMb. Hb, Hc are defined likewise.

The nodal tangents are perpendicular if and only if P lies on K168 = pK(X3, X2) in which case the node lies on the line OP.

The asymptotes concur if and only if P lies on the anticomplement of pK(X69, X2), a cubic passing through X(2), X(6) – see below – and X(6337).

When P is a point on the line at infinity, (C) passes through X(2) and the node is X(3). One of the nodal tangents is the perpendicular at X(3) to the line PX(3).

Examples :

When P is the Lemoine point X(6), (C) is the circumcircle and (K) decomposes into the perpendicular bisectors of ABC.

When P is the incenter X(1) of ABC, (K) becomes the circum-cubic K654 with node X(84). Three other similar circum-cubics are obtained with the excenters.

When P is the centroid X(2) of ABC, (C) is the Steiner ellipse and (K) is K461, a cubic with asymptotes concurring at X(382).


The isogonal transform of K654 is K926, a Lemoine cubic.