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X(2), X(4), X(7), X(8), X(20), X(69), X(189), X(253), X(329), X(1032), X(1034), X(5932), X(14361), X(14362), X(14365), X(34162), X(39158), X(39159), X(39160), X(39161), X(41080), X(42427), X(42428) X(55830) → X(55837) vertices of the 2nd Conway triangle, see ETC X(9776) vertices of antimedial triangle vertices of the Lucas triangle, see below foci of the Steiner circumellipse (see below) points at infinity of the Thomson cubic 

CPCC or Hcevian points, see Table 11, and their isotomic conjugates seven central cyclocevian points, see Table 24. See also Table 28 : cevian and anticevian points. *** Reference : Question 1207 raised by Édouard Lucas, Nouvelles annales de mathématiques 2e série, tome 15 (1876), p. 240. 

The Lucas cubic is the isotomic pK with pivot X(69), isotomic conjugate of H. It is the Thomson cubic of the antimedial triangle. See K878 for a generalization and other related cubics. Its isogonal transform is K172. It is a member of the classes CL023, CL024 of cubics. See also Table 15. It is anharmonically equivalent to the Thomson cubic. See Table 21. Locus properties :




Transformations on the Lucas cubic and associated group law The Lucas cubic is invariant under isotomic conjugation and under several other transformations. Its is indeed the locus of point P such that :
As in the pages K002 and K004, a group law is defined on K007 with neutral element X(69). Each center P on K007 is associated with an integer n. Three points are collinear if and only if the sum of the corresponding integers is 4. Q is the tangential of P. 





If P and P' are two points with integers n and n' then : • P and P' are X(69)Ceva conjugates hence collinear with X(4) if and only if n + n' = 0, • P and P' are agc conjugates hence collinear with X(2) if and only if n + n' = 2, • P and P' are isotomic conjugates hence collinear with X(69) if and only if n + n' = 4, • P and P' are cyclocevian conjugates hence collinear with X(20) if and only if n + n' = 6, • P and P' are X(4)cross conjugates hence collinear with X(14361) if and only if n + n' = 8. The tangential Q of P corresponds to 4  2n hence the isotomic conjugate of Q corresponds to 2n. For example, the point P(5) = X(34162) = X69/X189 = agcX1034 = tP(9). P(5) and P(9) are the anticomplements of X(3342) and X(3351). Similarly, P(8), P(7), P(10) and P(12) are the anticomplements of X(3349), X(3352), X(3356) and X(14481) respectively. 



The Lucas cubic and the foci of the Steiner ellipse The Lucas cubic is the most familiar example of isocubic which contains the (four) foci of the Steiner (circum) ellipse. See also K347 and K348. Indeed, these foci are the anticomplements of those of the inscribed Steiner ellipse which lie on the Thomson cubic since they are two isogonal conjugates collinear with the pivot G. Let then be F1, F2 the real foci and F1', F2' the imaginary foci of the Steiner ellipse, now X(39158), X(39159) and X(39160) X(39161) in ETC. Their isotomic conjugates tF1, tF2, tF1', tF2' obviously also lie on the Lucas cubic. Furthermore :
Naturally, all these results can easily be adapted to the imaginary foci. 

More informations in Wilson's page. See the related cubic K709. 



Generalization of the cyclocevian conjugation Let (Γ) be the circumconic with perspector M. Let P be a finite point with cevian triangle PaPbPc. There is one and only conic (C) passing through Pa, Pb, Pc and the infinite points of (Γ). (C) is a bicevian conic with perspectors P and another point Q we shall call the (Γ)cevian conjugate of P. With M = p : q : r and P = x : y : z, the first barycentric coordinate of the isotomic conjugate of Q is given by : – p / (x(y + z)) + q / (y(z + x)) + r / (z(x + y)) Note that (Γ) and (C) must meet again at two (real or not) finite points which lie on the trilinear polar of the barycentric product P x Q. Obviously, with M = X(6), (Γ) is the circumcircle of ABC and Q is the cyclocevian conjugate of P. Q = f(M, P) = ta(M ÷ ctP) where ÷ denotes a barycentric quotient and g, t, c, a denote an isogonal conjugate, an isotomic conjugate, a complement, an anticomplement as usual. In other words, if mX denotes the Misoconjugate of X, we obtain Q = tamctP. In particular, with M = X(6), we find Q = tagctP, as found a long time ago by Darij Grinberg (Hyacinthos #6423, January 24, 2003). See also Wilson's page mentioned above. Now, let (K) be the isotomic pivotal cubic pK(X2, S). For any point X on (K), the point Y = f(cS, X) also lies on (K) and the line XY passes through atS which can be seen as another pivot on the curve. atS is the PCeva conjugate of X(2). With S = X(69), (K) is the Lucas cubic K007, cS = X(6) and atS = X(20) which is property 5 above. 



Lucas triangle and Lucas isogonal conjugation K007 meets the circumcircle at A, B, C and three other points L1, L2, L3 on the rectangular hyperbola (H) passing through X(2), X(20), X(54), X(69), X(110), X(2574), X(2575), X(2979). These are the vertices of the Lucas triangle LT whose orthocenter is X(2979). LT is always an acutangle triangle. Its centroid is X = a^2 (a^6 b^23 a^4 b^4+3 a^2 b^6b^8+a^6 c^211 a^4 b^2 c^2+9 a^2 b^4 c^2+b^6 c^23 a^4 c^4+9 a^2 b^2 c^4+3 a^2 c^6+b^2 c^6c^8) : : , on the lines {3, 54}, {51, 631}, SEARCH = 7.07052700014795. X is obviously the image of X(2979) under the homothety h(O, 1/3). Note that (E) = {3, 54} is the Euler line of LT. X is now X(54041) in ETC. The Lemoine point of LT is X(54043). 

The conic (C) is inscribed in both triangles ABC and LT. Its center is X(3819) and its perspector is X(34384). *** Under isogonal conjugation with respect to LT, every line passing through O is transformed into a rectangular circumhyperbola of LT passing through X(2979). The hyperbola (H) above is the transform of the Euler line of ABC. The hyperbolas through {X22, X30, X476, X523, X2979} and {X329, X513, X517, X901, X2979} are the transforms of the lines through X(74), X(100) respectively. All these hyperbolas have their center on the nine points circle of LT, with radius R/2 and with center the midpoint of {X3,X2979}, on the lines {X3,X54}, {X51,X140}, {X143,X631}. This point is now X(54042) in ETC. *** 

pairs {P,Q} of isogonal conjugates with respect to LT : {2,20}, {3,2979}, {4,11206}, {8,9778}, {22,11459}, {54,52397}, {69,376}, {95,36987}, {329,5731}, {3534,38397}, {5890,20477}, {15531,21312}, {39158,39159}, {39160,39161}. Note that the last four points are the foci of the Steiner ellipse which lie on K007. For every point on the line at infinity or on the circumcircle, the isogonal conjugate with respect to LT coincide with the Thomson isogonal conjugate. *** Every pK(Ω, P) that passes through L1, L2, L3 must have • its pole Ω on psK(X54034, X95, X2) passing through X(2), X(6), X(97), X(4) × X(11206), X(22) × X(54), • its pivot P on psK(X95, X34384, X4) = spK(X51, X3819) passing through X(4), X(54), X(69), X(95) , X(2979), X(18018), X(275) × X(1370). Examples : K646 = pK(X97, X95), pK(X6, X2979). See also the related cubic K1326. 
