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K880

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X(4), X(9), X(72), X(226), X(329), X(1490), X(1903), X(2184), X(8804), X(8805), X(8806), X(8807), X(8894)

cevians of X(329)

vertices of the 2nd extouch triangle (not represented)

For any point M on the Lucas cubic K007, the cevian triangle of M is perspective to the 2nd [resp. 3rd] extouch triangle and the locus of the perspector is K880 = pK(X37, X329) [resp. pK(X1427, X5932)]. See ETC, preamble of X(8782).

K880 is the pK with pivot X(72), isopivot X(226) with respect to the 2nd extouch triangle (see ETC, X5927). These two points are in fact X(4), X(184) for this triangle with barycentric product X(32), hence K880 is K176 of the 2nd extouch triangle.

K880 meets the line at infinity at the same points as K343 = pK(X6, X63). Their six remaining common points A, B, C, X9 (twice), X2184 lie on a circum-conic (C) passing through X2, X9, X200, X281, X282, X346, X2184, X2287, X2297, etc. The three curves share the same tangent (L) at X(9), namely the line X(1), X(6), X(9), etc.

Hence K343 and K880 generate a pencil of cubics which contains the decomposed cubic into (L) and (C) and also a (not very interesting) central cubic.