   too complicated to be written here. Click on the link to download a text file.  X(2), X(6), X(83), X(3329), X(14617), X(41296), X(41297) Geometric properties :   K1183 is the locus of pivot of pK(Ω,P) passing through X(2), X(6), X(83). It is a nodal cubic with node X(83). The pole Ω must lie on the line (L) or on the conic (C). More precisely : • When Ω lies on the line (L) passing through X(2), X(32) and many other centers, the pivot must be X(83). (L) is the trilinear polar of X(4577). These cubics are in a same pencil containing K644 = pK(X251, X83). • When Ω lies on the circum-conic (C) with perspector X(5027), one can find one and only one associated pivot P on K1183. (C) passes through X(6), X(385), X(1691), X(3407) corresponding to the cubics K423, K739, K252, K421. K1183 passes through every center P(n) with first barycentric : (a^2 b^2 + a^2 c^2 + b^2 c^2 + a^4 (-1+n)) / (b^2 + c^2 + a^2 n) where n can be a real number or infinity and more generally any rational symmetric function f(a,b,c) with numerator and denominator of same degree. With P(n) ≠ X(83), the pole on (C) of the pK has first barycentric : a^2 (a^4 - b^2 c^2) / (b^2 + c^2 + a^2 n). For example P(∞) = X(6), P(1) = X(2). With n = – (a^4+a^2 b^2+b^4+a^2 c^2+b^2 c^2+c^4)/(a^2 b^2+a^2 c^2+b^2 c^2), we find P(n) = X(3329). With n = – (a^4 b^4+a^4 b^2 c^2+a^2 b^4 c^2+a^4 c^4+a^2 b^2 c^4+b^4 c^4)/(a^2 b^2 c^2 (a^2+b^2+c^2)), we find P(n) = X(14617). Obviously, two special different values of n give the same point X(83), the node of the cubic. These are the same as the ones we found in K1182, namely : 1 ± √(a^2 + b^2 + c^2) √(a^2 b^2 + a^2 c^2 + b^2 c^2) / (a b c). *** Let F be the quadratic transformation denoted BGF in ETC, introduced in the preamble just before X(41231). See also CL073. Its inverse is F-1 or invBGF. With X = x:y:z, we have F(X) = b^4 c^2 x^2-b^2 c^4 x^2-a^4 c^2 x y+b^2 c^4 x y+a^4 b^2 x z-b^4 c^2 x z-a^4 b^2 y z+a^4 c^2 y z : : . F-1(X) = a^2 (b^4 x^2-c^4 x^2-a^2 b^2 x y+c^4 x y-b^4 x z+a^2 c^2 x z+a^2 b^2 y z-a^2 c^2 y z) : : . F(X) is the intersection of the lines X(2), X and X(6), tgX = isotomic of isogonal of X. F has three singular points namely X(2), X(6), X(32). It fixes A, B, C and more generally, every point on the circum-conic with perspector X(669). F transforms every pK passing through these three singular points (the pKs mentioned in K1182) into another pK passing through X(2), X(6), X(83) i.e. those mentioned above. Note that the pivot of the former pK is the pole of the latter one unless one of them is a member of a pencil mentioned in K1182 and K1183. For example, K1016 = pK(X18899, X6) is transformed into K423 = pK(X6, X3329). F transforms K1182 into (C) and (C) into K1183 hence F^2 maps K1182 onto K1183. The singular points X(2), X(6), X(32) on K1182 must be excluded. F^2 is another quadratic transformation with singular points X(2) and X(32) which is double. F^2 maps every point on the line X(32)X(2) onto X(83) and every point on the line X(32)X(1501) onto X(2). The fixed points are those (apart X32) on the circum-conic with perspector X(669), passing though X(i) for i = 6, 32, 83, 213, 729, 981, 1918, 1974, 2207, 2281, 2422, 3114, 3224, 3225, etc. Recall that these fixed points are those of F and F-1. F and F-1 are easily used to find lots of centers on K1182 and K1183. Indeed, if M is on the line at infinity, the barycentric quotient N = X(5027) ÷ M is on (C) hence F(N) is on K1183 and F-1(N) is on K1182. The following table shows a selection of such points. M F-1(N) on K1182 SEARCH X512 X6 0.9929084950656698 X513 a^4 (b+c) (a^3 b^2+a^3 b c+a^2 b^2 c+a b^3 c+b^4 c+a^3 c^2+a^2 b c^2+a b^2 c^2+b^3 c^2+a b c^3+b^2 c^3+b c^4) 0.5416784695870229 X514 a^6 (b+c) (a^3 b^3+a^3 b^2 c+a^2 b^3 c+a^3 b c^2+a^2 b^2 c^2+a b^3 c^2+b^4 c^2+a^3 c^3+a^2 b c^3+a b^2 c^3+b^3 c^3+b^2 c^4) 0.1398216813933225 X523 X1501 0.06809137682451109 X525 a^6 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2+a^2 c^2-b^2 c^2) 0.0053029394200887 X688 X2 2.629368792488718 X690 a^6 (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (a^2 b^2+a^2 c^2-2 b^2 c^2) 0.1493261025922344 X732 a^6 (b-c) (b+c) (a^4 b^4+a^2 b^6-a^4 b^2 c^2-2 a^2 b^4 c^2+b^6 c^2+a^4 c^4-2 a^2 b^2 c^4-b^4 c^4+a^2 c^6+b^2 c^6) -2.272850939196628 X740 a^6 (b-c) (a^2 b^4-a^2 b^3 c-a b^4 c+b^5 c+a^2 b^2 c^2-b^4 c^2-a^2 b c^3+b^3 c^3+a^2 c^4-a b c^4-b^2 c^4+b c^5) -2.538016294733149 X782 a^6 (b^4+a^2 c^2) (a^2 b^2+c^4) (a^4 b^4+a^2 b^6+b^6 c^2+a^4 c^4+a^2 c^6+b^2 c^6) 0.2016303971683785 X812 a^6 (b+c) (a^2 b^4+a^2 b^3 c+a b^4 c+b^5 c+a^2 b^2 c^2+b^4 c^2+a^2 b c^3+b^3 c^3+a^2 c^4+a b c^4+b^2 c^4+b c^5) 0.2582622512638695 X826 X9233 0.01539761869359225 X888 a^2 (2 a^2-b^2-c^2) (2 a^2 b^2-a^2 c^2-b^2 c^2) (a^2 b^2-2 a^2 c^2+b^2 c^2) 1.226347176040271 X1499 a^6 (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (a^2 b^2+a^2 c^2-5 b^2 c^2) 8.369408051760912 X3221 X3504 10.71386048162398 M F(N) on K1183 SEARCH X512 X6 0.9929084950656698 X513 (b+c) (a^6+a^5 b+a^4 b^2+a^3 b^3+a^5 c+a^3 b^2 c+a^2 b^3 c+a^4 c^2+a^3 b c^2+a^3 c^3+a^2 b c^3+b^3 c^3) 1.454000928403231 X514 (b+c) (a^6+a^4 b^2+a^3 b^3+a^3 b^2 c+a^2 b^3 c+a^4 c^2+a^3 b c^2+a b^3 c^2+a^3 c^3+a^2 b c^3+a b^2 c^3+b^3 c^3) 2.164323516817754 X523 X2 2.629368792488718 X525 (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^4-a^2 b^2-a^2 c^2-b^2 c^2) 0.8890402050649093 X688 b^2 (a^2+b^2) c^2 (a^2+c^2) (a^4+a^2 b^2+a^2 c^2-b^2 c^2) -1.00851874620026 X690 (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (3 a^4-a^2 b^2-a^2 c^2-b^2 c^2) 1.808992796849457 X732 (a^2+b^2) (b-c) (b+c) (a^2+c^2) (a^8 b^2+a^6 b^4+a^8 c^2-4 a^4 b^4 c^2+a^2 b^6 c^2+a^6 c^4-4 a^4 b^2 c^4+b^6 c^4+a^2 b^2 c^6+b^4 c^6) 2.92685624085448 X740 (b-c) (a^6-a^5 b+2 a^4 b^2-a^3 b^3-a^5 c-2 a^3 b^2 c+2 a^2 b^3 c+2 a^4 c^2-2 a^3 b c^2-a b^3 c^2-a^3 c^3+2 a^2 b c^3-a b^2 c^3+b^3 c^3) 2.908241758378135 X782 (a^2+b^2) (a^2+c^2) (b^4+a^2 c^2) (a^2 b^2+c^4) (a^8 b^2+2 a^6 b^4+a^4 b^6+a^8 c^2+3 a^6 b^2 c^2+2 a^4 b^4 c^2+2 a^2 b^6 c^2+2 a^6 c^4+2 a^4 b^2 c^4+3 a^2 b^4 c^4+b^6 c^4+a^4 c^6+2 a^2 b^2 c^6+b^4 c^6) 1.620917072373061 X812 (b+c) (a^6+a^5 b+2 a^4 b^2+a^3 b^3+a^5 c+2 a^4 b c+2 a^3 b^2 c+2 a^2 b^3 c+2 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2+a b^3 c^2+a^3 c^3+2 a^2 b c^3+a b^2 c^3+b^3 c^3) 1.766062549547471 X826 (a^2+b^2) (a^2+c^2) (a^4-a^2 b^2-a^2 c^2-b^2 c^2) 5.918095608809818 X888 (a^2 b^2-2 a^2 c^2+b^2 c^2) (-2 a^2 b^2+a^2 c^2+b^2 c^2) (-a^4-a^2 b^2-a^2 c^2+3 b^2 c^2) 2.349486871120276 X1499 (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (6 a^4-a^2 b^2-a^2 c^2-b^2 c^2) 3.356615593506579 X3221 (a^2 b^2-a^2 c^2+b^2 c^2) (-a^2 b^2+a^2 c^2+b^2 c^2) (-a^4-a^2 b^2-a^2 c^2+2 b^2 c^2) 3.985809557529503  