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See the related K397 for another equation See also K584 for a trilinear equation and related comments 

X(1), X(3), X(4), X(5), X(17), X(18), X(54), X(61), X(62), X(195), X(627), X(628), X(2120), X(2121), X(3336), X(3459), X(3460), X(3461), X(3462), X(3463), X(3467), X(3468), X(3469), X(3470), X(3471), X(3489), X(3490), X(6191), X(6192), X(7344), X(7345), X(8837), X(8839), X(8918), X(8919), X(8929), X(8930), see table below Ia, Ib, Ic excenters 

Ha, Hb, Hc projections of G on the altitudes Oa, Ob, Oc their isogonal conjugates centers of the 6 equilateral triangles erected on the sides of ABC, externally (Ae, Be, Ce) or internally (Ai, Bi, Ci), see figure below centers of equilateral cevian and precevian (anticevian) triangles (JeanPierre Ehrmann). See table 10, table 14. Ixanticevian points and their isogonal conjugates : see table 23 four foci of the MacBeath inconic : X(3), X(4) and two imaginary X(5)circumcevian points (see Q141) *** 

The NapoleonFeuerbach cubic is the isogonal pK with pivot X(5) = ninepoint center. See Table 27. It is anharmonically equivalent to the Neuberg cubic, see Table 20. The Psi transform of K005 is Q041. The symbolic substitution SS{a > √a} transforms K005 into K1081. 

Locus properties : (see also Z. Cerin's paper in the bibliography)


Points on the Napoleon cubic 

Altitudes and cevians of O K005 contains : – Ha, Hb, Hc projections of G on the altitudes of ABC – #O = X(3470), the common tangential of O and Ha, Hb, Hc – Oa, Ob, Oc isogonal conjugates of Ha, Hb, Hc, these points on the cevians of O – #H, the common tangential of H and Oa, Ob, Oc Note that the tangents at O and H pass through X(74) and that #O, #H and X(54) are collinear (they are the tangentials of O, H and X(5)) on the satellite line of the Euler line. – (#O)* = X(3471) and (#H)*, isogonal conjugates of #O and #H 

Napoleon triangles K005 contains : – the vertices of the outer triangle AeBeCe. The tangents at Ae, Be, Ce, X61 concur at #X61 on the curve. – the vertices of the inner triangle AiBiCi. The tangents at Ai, Bi, Ci, X62 concur at #X62 on the curve. – the third points Ae', Be', Ce' on the sidelines of AeBeCe (these two triangles are perspective at X61). – the third points Ai', Bi', Ci' on the sidelines of AiBiCi (these two triangles are perspective at X62). Note that #O is also the perspector of the triangles Ae'Be'Ce' and Ai'Bi'Ci' and that #O, #X61, #X62 are collinear on the satellite line of the Brocard line. 

Centers of the equilateral cevian or anticevian triangles K005 contains the six centers (not always real as seen in the figure) of the equilateral cevian points i.e. the points (such as X(370)) whose cevian triangle is equilateral. These equilateral cevian points lie on the Neuberg cubic. These centers lie on the perpendicular bisectors of X(13)X(15) (red points) and X(14)X(16) (blue points). See Table 10 for more details.
K005 also contains the six centers (not always real) of the equilateral anticevian points i.e. the points whose anticevian triangle is equilateral. See Table 14 for more details and a figure.
This gives 24 points on K005, counting the isogonal conjugates. 

Ixanticevian points K005 contains the four (green points, not always real) Ixanticevian points. These points are the common points (apart A, B, C, H, X5) of the cubics of the pencil which contains K005, K049, K060 and many other cubics. See Table 23 for explanations. Obviously K005 also contains the isogonal conjugates (purple points) of these four points. This gives eight more points on the cubic. 

The table gives centers on the cubic. Each weak point (in red) must have its three extraversions on the curve. The SEARCH value is given for these points. Most of the points are the perspectors of two inscribed triangles in the cubic. The Avertex is given in the table. P* is the isogonal conjugate of P and P/Q is the cevian quotient of P and Q. 






