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X(2), X(5), X(32), X(51), X(132), X(206), X(216), X(232), X(237), X(511), X(3613), X(11672), X(40588), X(40601), X(52878), X(52967), X(60517)

X(52967) = X(51) x X(511) on the lines {X5,X53}, {X32,X206}, {X132,X232}, etc

X(60524) → X(60528), X(60595), X(60596)

medial triangle

infinite points of the circum-conic with perspector X(51)

Geometric properties :

K1355 is the barycentric product X(511) x K1354. Its isogonal transform is K1424.

These two cubics K1354, K1355 share the same points at infinity and meet again at A, B, C, X(2), X(51) and a sixth point which is the barycentric product P0 = X(5) x X(98) on the lines {X2,X290}, {X4,X32}, {X5,X217}, etc.

P0 = (a^4+b^4-a^2 c^2-b^2 c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-a^2 b^2-b^2 c^2+c^4) : : , SEARCH = -0.249009163063951. P0 is now X(60517) in ETC.

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Other centers on K1355

P1 = (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : : , SEARCH = -1.34066116913094

P2 = a^4 (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^6-a^4 b^2-a^2 b^4+b^6+a^2 c^4+b^2 c^4-2 c^6) (a^6+a^2 b^4-2 b^6-a^4 c^2+b^4 c^2-a^2 c^4+c^6) : : , SEARCH = 0.296723043496870

P3 = (a^4+b^4-c^4) (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-b^4+c^4) : : , SEARCH = 0.723977632843009

P4 = (a^4+a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^2 b^2-b^4+a^2 c^2-c^4) (a^4-a^2 b^2+a^2 c^2-b^2 c^2+c^4) : : , SEARCH = 1.94176536246465

P5 = a^4 (a^4-a^2 b^2-a^2 c^2-b^2 c^2) (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : : , SEARCH = 0.205847456435866

P6 = a^2 (a^2 b^2-b^4+a^2 c^2+b^2 c^2) (a^2 b^2+a^2 c^2+b^2 c^2-c^4) (a^4 b^2-a^2 b^4+a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4) : : , SEARCH = 0.862578830445220

P7 = a^2 (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^6-2 a^4 b^2+a^2 b^4-2 a^4 c^2-a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4) : : , SEARCH = 1.22262111656355