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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.
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.