X(1), X(3), X(4), X(13), X(14), X(15), X(16), X(30), X(74), X(370), X(399), X(484), X(616), X(617), X(1138), X(1157), X(1263), X(1276), X(1277), X(1337), X(1338), X(2132), X(2133), X(3065), X(3440), X(3441), X(3464), X(3465), X(3466), X(3479), X(3480), X(3481), X(3482), X(3483), X(3484)
excenters; reflections of A, B, C in the sidelines of ABC; cevians of X(30)
vertices of 6 equilateral triangles erected on the sides of ABC
See table 19 for a description of these centers and more points on the Neuberg cubic.
The Neuberg cubic is introduced in Neuberg's paper "Mémoire sur le tétračdre" in Mémoires de l’Académie de Belgique, pp.1–70, 1884. The characterization given is related with properties of so-called "quadrangles involutifs" :
The Neuberg cubic is the isogonal pK with pivot X(30) = infinite point of the Euler line : it is the locus of point P such that the line PP* is parallel to the Euler line (P* isogonal conjugate of P). See Table 27. It is a circular cubic with singular focus X(110), focus of the Kiepert parabola. It is also the orthopivotal cubic O(X3) and C(0) = C(infty) in "On two Remarkable Pencils of Cubics of the Triangle Plane" (see Downloads page). It is sometimes called 21-point cubic or 37-point cubic in older literature.
See Table 20 for cubics anharmonically equivalent to the Neuberg cubic.
Locus properties : (see also Z. Cerin's papers in bibliography)
Other properties :
See details in Special Isocubics, §6.5 and also table 18.
Ga, Gb, Gc are the vertices of the antimedial triangle. Denote by (Ha) the hyperbola passing through B, C, Ga, the reflection A' of A in the line BC, the reflection Ah of H with respect to the second intersection of the altitude AH with the circumcircle. The asymptotes of (Ha) make 60° angles with the sideline BC. (Hb) and (Hc) are defined similarly. See figure 1.
(Ha), (Hb), (Hc) have three points Ua, Ub, Uc in common which lie :
The orthocenter of triangle UaUbUc is O and its circumcenter is L. Thus ABC and UaUbUc share the same Euler line.
All rectangular hyperbolas passing through Ua, Ub, Uc, O are centered on the circle centered at X(550) with radius R (X(550) is the reflection of the nine point center X(5) in the circumcenter O).
The isogonal conjugates Ua*, Ub*, Uc* of Ua, Ub, Uc , the midpoints Va, Vb, Vc of triangle Ua*Ub*Uc*, their isogonal conjugates Va*, Vb*, Vc* are nine other points on the Neuberg cubic. Notice that Va, Vb, Vc are the complements of Ua*, Ub*, Uc* respectively, these latter points lying on the circle C(O,2R). Thus the translation with vector OH maps the triangle UaUbUc to the triangle Ua*Ub*Uc*. Note that the lines Ua*Va*, Ub*Vb*, Uc*Vc* concur at X(399), the Parry reflection point. See figure 3.
Hence we know the nine common points of the Neuberg cubic and the cubic which is its anticomplement : these are O, the Fermat points X(13) and X(14), the point at infinity X(30) of the Euler line, the three points Va, Vb, Vc and the circular points at infinity.
At last, remark that all the cubics passing through A, B, C, Ga, Gb, Gc, Ua, Ub, Uc are equilateral cubics and the equilateral triangle formed by the asymptotes has always center O. The only K60+ is K0++ = K080 which is at the same time a central cubic.
See also the related cubic K405 which contains Ua*, Ub*, Uc*.
Under the rotations with center O (resp. H) and angles +/-2π/3, the Neuberg cubic is transformed into two other circular cubics K001+ and K001–. These two cubics generate a pencil of circular cubics passing through O (resp. H) and therefore having six other common points. This pencil contains the Neuberg cubic itself hence there are two equilateral triangles with center O (resp. H) inscribed in the Neuberg cubic. See the two figures below.
A line passing through O meets the Neuberg cubic again at two points M, N (see property 8 above). The midpoint P of MN lies on the Stammler strophoid K038. Note that these points M, N lie on a same circum-conic passing through X(1138).
Similarly, a line passing through H meets the Neuberg cubic again at two points M, N and their midpoint lies on another strophoid K591 we shall call the Kiepert-Neuberg strophoid.
The osculating circle (Ca) at A to the Neuberg cubic passes through A, the traces of X(30) and X(526) on the sideline BC. Its center Oa lies on the line AX(110).
(Cb), (Cc) and their centers Ob, Oc are defined similarly.
The triangle OaObOc is perspective with
• ABC at X(110),
• the cevian triangle of X(476) at X(523),
• the anticevian triangle of any point on the cubic K130 = pK(X6, X476).
The locus of the perspector is the isogonal pK with pivot P = X(5)X(1117) /\ X(110)X(476) passing through the in/excenters, X(110), X(477), X(523) and Oa, Ob, Oc.
Note that this cubic is tangent at X(523) to the line at infinity and tangent at X(110) to the circumcircle.
See figure below.
The third points Ua, Ub, Uc on the sidelines of OaObOc are the traces of X(523).
The third points Va, Vb, Vc on the sidelines of UaUbUc are the traces of the isogonal conjugate of X(477).
The perspectors of any two triangles ABC, IaIbIc, OaObOc, UaUbUc, VaVbVc and the cevian triangle of P are centers on the cubic.
The last two common points with the Neuberg cubic are E1, E2 on the parallel at P to the Euler line.
This cubic is also a pK with respect to OaObOc with pivot X(523), isopivot the infinite point of the line X(3)X(74)X(110).
Let M be a point and C(M) its polar conic.
• a circle when M = X(110), the singular focus,
• a rectangular hyperbola when M lies on the Euler line, the orthic line of the cubic,
• a parabola when M lies on (P), the poloconic of the line at infinity,
• an ellipse when M lies inside the (light blue) region that contains X(110),
• a hyperbola when M lies outside this same region.
(P) is a very remarkable hyperbola passing through X(476) and the vertices of the circumtangential triangle. It has two asymptotes making an angle of 60° thus its eccentricity is 2.
X(110) is one of its foci and the related directrix is the Euler line.
The tangent at X(476) is the real asymptote of the Neuberg cubic.
For any point P on the Neuberg cubic, recall that the Euler lines (Ea), (Eb), (Ec) of triangles PBC, PCA, PAB concur at M on the Euler line (E) of ABC.
Conversely, for a given point M on (E), we seek points on the Neuberg cubic having this property.
There are two such points P, Q on K001 which are the common points of the rectangular circum-hyperbola (H) through M and the line (L) passing through X(399) and the reflection of O in M. Thus, the Euler lines of the six triangles PBC, PCA, PAB, QBC, QCA, QAB concur at M on (E).
It follows that K001 can be seen as the locus of the intersections of (H) and (L) when M traverses (E).
Similarly, for a given point M on the Brocard axis, there are three points P1, P2, P3 such that the Brocard lines of the nine analogous triangles concur at M.
More details here.