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X(1), X(3), X(4), X(20), X(40), X(64), X(84), X(1490), X(1498), X(2130), X(2131), X(3182), X(3183), X(3345), X(3346), X(3347), X(3348), X(3353), X(3354), X(3355), X(3472), X(3473), X(3637) antipodes of A, B, C on the circumcircle excenters, infinite points of the altitudes CPCC or Hcevian points, see Table 11, their reflections about X(3), their isogonal conjugates foci of the inconic with center X(3), X1OAP points, see K033, K806, K807 and also Table 53 other points below 

The Darboux cubic is the isogonal pK with pivot L = X(20). See Table 27. It is the only isogonal central pK. Its center is O. Hence, it is a pK++. See also table 15. It is anharmonically equivalent to the Thomson cubic. See Table 21. K004 is a member of the class CL043 : it meets the circumcircle at A, B, C and at their antipodes where the tangents are concurrent at the point X(1498). See also Q063. The isotomic transform of K004 is K183. K004 is its own sister as in Table 58. Locus properties (see also Table 6)


Other properties (Ha) is the hyperbola centered at A, passing through the antipode Ao of A in the circumcircle, through its reflection A1 in A and whose asymptotes are the perpendiculars at A to the sidelines AB and AC (these are the lines ABo and ACo). (Ha) meets the circle C(O,3R) at four points A1 and U, V, W. Naturally, U, V, W also lie on two similar hyperbolas (Hb) and (Hc) and the isogonal conjugates U*, V*, W* of U, V, W lie on K004. U, V, W and their reflections U', V', W' in O are six points on the Darboux cubic. Furthermore, the triangle UVW has orthocenter L, centroid X(376) (the reflection of G in O) and the midpoints U", V", W" also lie on K004. They are the isogonal conjugates of U', V', W' and the midpoints of LU', LV', LW'. Obviously, the reflections U3, V3, W3 of U", V", W" about X(3) also lie on K004. 

U', V', W', A1, B1, C1 are the six points where C(O,3R) meets (T), the homothetic of the Thomson cubic under h(O,3), thus they are the images of the vertices of the Thomson triangle under the same homothety. U', V', W' also lie on three rectangular hyperbolas (H'a), (H'b), (H'c) passing through H. (H'a) contains A, the reflection of L in A and its asymptotes are parallel to the bisectors at A. Its center is the homothetic of L under h(A,1/2). Its tangent at A is the line AX(64) i.e. it is tangent at A to K004. Most of these points are directly related with the vertices Q1, Q2, Q3 of the Thomson triangle : U, U', U", U3 are the images of Q1 under h(X3, 3), h(X3, 3), h(X4, 3/2), h(X631, 3/2) respectively. The other points are defined similarly. 



A remarkable pencil of cubics The Darboux cubic K004 and the decomposed cubic which is the union of the circumcircle and the Euler line generate a pencil of central cubics with center O passing through H, X(20) and the reflections A', B', C' of A, B, C about O. Each cubic is spK(P, Q) in CL055 where P is a point on the Euler line and Q is the midpoint of PX(20). spK(P,Q) contains the infinite points of pK(X6, P) and the isogonal conjugate of P. This pencil also contains (apart K004 which is the only pK) the cubics K047, K080, K426, K443, K566 corresponding to K002, K003, pK(X6, X3146), K006, K005 respectively. Note that K426 and K443 are psK cubics. See PseudoPivotal Cubics and Poristic Triangles. 



K004 and the foci of the Steiner inellipse The Darboux cubic K004 is the homothetic of K758 under h(X3, 3). Since this latter cubic contains the four foci of the Steiner inellipse, K004 must contain their images under the same homothety. h(X4, 3) transforms these same foci into four other points, also on K004, which are the reflections about X(3) of the four previous points. 



Group structure and organization of points on the Darboux cubic The usual group structure is easily adapted on K004 : the sum M+N of two points is the isogonal conjugate of the third intersection S of K004 with the line MN and, obviously, X(20), S, M+N are collinear. The pivot X(20) is the neutral element 0 of the group. Following Fred Lang (Forum Geometricorum, vol.2, 2002, pp.135146), we construct the following table (splitted into a negative part and a positive part) taking into account that K004 is stable under four transformations (detailed underneath) namely isogonal conjugation, reflection about X(3), X(30)Ceva conjugation, X(64)crossconjugation. 





Each point P on K004 corresponds to an integer n and Q = tg(P) denotes the tangential of P. Three points on K004 are collinear if and only if their sum is 6. This gives the folllowing assertions generalized below. • the sum of two isogonal conjugates P, P* is 6 since they are collinear with the neutral element X(20). • the sum of two points P, P' symmetric about X(3) is 4 since X(3) is 2. Equivalently, the arithmetic mean is 2. • the sum of P and its X(30)Ceva conjugate X30/P is 0 since they are collinear with X(64), the tangential of X(20) with value 6. • the sum of P and its X(64)cross conjugate X64©P is 12 since they are collinear with X(2130), the tangential of X(64) with value 6. The lefthand table shows a selection of points deduced from P with their corresponding integer. The righthand table shows pairs of points with sum s hence collinear with a third point S of K004. 




Remark : tg(P') = tg(P)' with same number 2 + 2n which is obvious geometrically. 

In other words, this gives a quick and easy method to find the third point of K004 on a line passing through two given points. For example, the third point P on X(3183)X(3473) is given by 4+13+ n = 6 hence n = 3 so P = X(3182). Similarly, it is easy to identify the tangential of any point on K004. 
