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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), X(38931), X(38932), X(38933), X(38934), X(38935), X(39261), X(39262), X(46035), X(46036), X(46037), X(47304), X(47305), X(47306), X(47307), X(48795), X(48797), X(55483), X(55484), X(55485), X(55486), X(55487), X(55488), X(55489), X(55490), X(55491), X(55492), X(55493), X(55494), X(55495), see table below

Ia, Ib, Ic excenters

four foci of the MacBeath inconic : X(3), X(4) and two imaginary

Ha, Hb, Hc projections of G on the altitudes and their isogonal conjugates Oa, Ob, Oc. These points are the vertices of the orthocentroidal and orthocentroidal-isogonic triangles. They also lie on K397.

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 (Jean-Pierre Ehrmann). See table 10, table 14.

Ix-anticevian points and their isogonal conjugates : see table 23

X(5)-circumcevian points (see Q141)

***

The Napoleon-Feuerbach cubic is the isogonal pK with pivot X(5) = nine-point center. See Table 27.

It is anharmonically equivalent to the Neuberg cubic, see Table 20. Its isotomic transform is pK(X76, X311) and its Psi transform is Q041.

The symbolic substitution SS{a -> √a} transforms K005 into K1081.

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

1. Denote by PaPbPc the pedal triangle of point P. The Euler lines of APbPc, BPcPa, CPaPb concur (at Q) if and only if P lies on the Napoleon cubic (together with the line at infinity). H, P, Q are collinear and the locus of Q is Q093. Compare with K001, property 5.
2. Under the homothety h(P,-2), PaPbPc is transformed into QaQbQc which is perspective to ABC if and only if P lies on the Napoleon cubic. For this reason, the Napoleon cubic is called -2-pedal cubic in Pinkernell's paper. The perspector lies on K060 which is the 2-cevian cubic.
3. Pa, Pb, Pc are the reflections of point P in the sidelines of triangle ABC. Oa, Ob, Oc are the circumcenters of triangles PaBC, PbCA, PcAB. The triangles ABC and OaObOc are perspective if and only if P lies on K060 (together with the circumcircle and the line at infinity). The locus of the perspector is the Napoleon cubic.
4. The cevian lines of point P meet the perpendicular bisectors of ABC at A', B', C'. The locus of P such that the sum of oriented lines angles (BA',BC)+(CB',CA)+(AC',AB) = pi/2 (mod. pi) is the Napoleon cubic. When this sum is 0 (mod. pi), we obtain the Brocard (second) cubic K018. (Jean-Pierre Ehrmann)
5. Let Q be a fixed point, QaQbQc its pedal triangle and P a variable point. The parallel to QbQc through P intersects AB and AC at Ab and Ac. A1 (resp. A2) is the apex of the equilateral triangle erected outwardly (resp. inwardly) on AbAc. Similarly define B1, B2 and C1, C2. The two triangles ABC, A1B1C1 (resp. A2B2C2) are perspective for all P if and only if Q lies on the Napoleon cubic. The perspector lies on K129b (resp. K129a). (Problem raised by Antreas Hatzipolakis, Hyacinthos #10560)
6. An orthopivotal cubic O(P) is a pK if and only if its orthopivot P lies on the Napoleon cubic. The singular focus of O(P) lies on Q041, the Psi-transform of K005.
7. Let P be a variable point. Reflect B in PC, C in PB and denote by La the perpendicular bisector of these two points. Lb, Lc are defined cyclically. La, Lb, Lc concur (at Q) if and only if P lies on K005, together with the circumcircle and the line at infinity. (Ngo Quang Duong, ADGEOM #3309). The locus of Q is Q114, together with the Euler line and the Jerabek hyperbola.
8. K005 is the Laplacian of the quintic Q003 i.e. the locus of points whose polar conic in the quintic is a rectangular hyperbola.
9. Let P be a point. Denote by Bc and Cb the reflections of B in the line PC and C in the line PB respectively. (Ca) is the circumcircle of triangle ABcCb and (Cb), (Cc) are defined likewise. These three circles are congruent if and only if P lies on K005 and then, their radical center is P. For P = X(3), these three circles coincide in the circumcircle of ABC (Kadir Altintas, 2021-01-12).
10. Let PaPbPc be the anticevian triangle of a point P and denote by Qa, Qb, Qc the reflections of Pa, Pb, Pc in the sidelines of ABC respectively. ABC and QaQbQc are perspective (at Q) if and only if P lies on K005. The locus of Q is K060. (Angel Montesdeoca, 2021-08-02). Note that PaPbPc and QaQbQc are perspective (at Q') if and only if P lies on K002. The locus of Q' is K004.
11. Let PaPbPc be the pedal triangle of a point P. Denote by Xa the point X(74) of triangle APbPc and define Xb, Xc cyclically. P, Xa, Xb, Xc lie on a same circle if and only if P lies on K005 (Kadir Altintas, 2021-07-24).
12. Let P be a point. Denote by Ga the centroid of PBC and by Ab, Ac the reflections of Ga in the cevians BP, CP resp. Define Ba, Bc, Ca, Cb cyclically. The circles (Ga, Bc, Cb), (Gb, Ac, Ca), (Gc, Ba, Ab) are congruent if and only if P lies on K005, and then, their radical center is P. For P = X(5), these three circles coincide in one circle centered at X(5), with radius R/3, hence having 9 points on it (Kadir Altintas, 2024-02-18, see Euclid #6131).
13. Let P be a point and let PaPbPc be its anticevian triangle. Denote by Oa, Ob, Oc the circumcenters of the triangles PPbPc, PPcPa, PPaPb. The triangles PaPbPc and OaObOc are perspective (at Q) if and only if P lies on a circular sextic S (together with the polar circle, the line at infinity and the sidelines of ABC). When P traverses S, the locus of Q is K005. Note that S is the square of K095. (Angel Montesdeoca, 2024-02-27)

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 = X(38933), 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.

Ix-anticevian points

K005 contains the four (green points, not always real) Ix-anticevian 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 A-vertex is given in the table.

P* is the isogonal conjugate of P and P/Q is the cevian quotient of P and Q.