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X(2), X(4), X(145), X(263), X(957), X(1992), X(5967), X(9214), X(12848), X(21463), X(21464), X(21466), X(21467), X(34244), X(40819), X(41316), X(41518), X(41519), X(42287), X(52765), X(55937), X(56850), X(63170),

X(63851) → X(63859), X(64973)

vertices of the pedal triangle of G

points at infinity of the Thomson cubic

common points of the circumcircle and pK(X6, X5640) where X5640 = X2-X51 /\ X6-X110

projections of G on the altitudes i.e. vertices of the orthocentroidal triangle, these points also on K005 and K397

Let M be the Miquel point of the quadrilateral formed by the sidelines of ABC and the trilinear polar L of P. Let M' be the second intersection of the line KM with the circumcircle. The Simson line of M' is perpendicular to L if and only if P lies on K015 (after Philippe Deléham). These lines are parallel if and only if P lies on K295.

K295 is the Lemoine generalized cubic K(X2) hence it is a nodal cubic with node G. See also Table 64 and CL045 where a construction is given.

The nodal tangents are parallel to the asymptotes of the Kiepert hyperbola.

It has three real asymptotes parallel to those of the Thomson cubic.

It meets the Steiner ellipse again at three points and the tangents at these points are concurrent.

The isogonal transform of K295 is K297.

Locus properties :

1. K295 = TC(X2) is a member of the class CL040 of Thomson centroidal cubics. It is the locus of point M such that the trilinear polar of M is perpendicular to the line GM.
2. K295 is a tripolar centroidal cubic of the class CL045. It is the locus of M whose tripolar centroid lies on the orthic axis. See the related K1360.
3. K295 is the locus of the orthopivots of the orthopivotal cubics that meet the sidelines of ABC again at three points which are the vertices of a pedal triangle. This pedal triangle is that of a point on the Kiepert hyperbola. See K059 and K452 for example.
4. see Q012, property 3 for another generalization.
5. (consequence of properties 1 and 2) Two points M, N on K295 and on a line passing through X(1992) share the same tripolar centroid T which lies on the orthic axis. In addition, the trilinear polars L(M), L(N) of M and N are perpendicular, and so are the lines GM and GN. See a construction below.

A construction

A variable line (L) passing through X(69) is reflected about G into the line (L') passing through X(1992).

The isotomic transform of (L) is a rectangular circum-hyperbola (H).

(L') and (H) meet at the requested points M, N with same tripolar centroid T.

Note that (L) and (H) meet on the Lucas cubic K007.

***

A selection of centers {M, N, T} by Peter Moses (last update 2024-11-15)

{4, 9214, 1637}, {145, ?, 14425}, {263, 52765, 2491}, {957, 63852, 3310}, {1992, 63854, 9125}, {5967, 63858, ?}, {41518, 41519, 647}, {42287, 63856, ?}, {55937, 63851, 676}