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K1077

a (–a + b + c) (y – z) y z = 0

X(1), X(2), X(8), X(9), X(188), X(236), X(3161), X(7028), X(8056), X(24150), X(24151), X(24152), X(24153), X(24154), X(24155), X(24156), X(24157), X(24158)

midpoints

vertices of the anticevian triangle of X(188) i.e. square roots of X(9)

points of (K) = pK(X2, X9) on the Steiner ellipse (S)

vertices of the second Zaniah triangle, see ETC X(18214)

Geometric properties :

K1077 is the image of the strong cubic K168 = pK(X3, X2) under the symbolic substitution SS{a -> √a}.

Its isogonal, isotomic, anticomplement are pK(X604, X57), pK(X85, X85), K1078 = pK(X2, X7) respectively.

K1077 is the barycentric products X(8) x K365 and X(75) x K761.

For any Ω on K1077, one can find a pK(Ω, P) having the same asymptotic directions as K1077 itself and then P lies on K1078. Examples of such pairings {Ω, P} = {X(i), X(j)} for these {i, j} : {1, 145}, {2, 7}, {8 , 8}, {9, 2}, {188, 7057}, {3161, 8055}, {7028, 7048}, {8056, 4373}.

All the cubics in {K201, K365, K747, K748, K761, K1077, K1078, K1079, K1082} are anharmonically equivalent.

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Remarkable polar conics PC(P) of P on K1077:

• PC(X1) is the complement of the Feuerbach hyperbola with center X(3035). It is the bicevian conic C(X2, X100) passing through X(i) for these i : 1, 3, 9, 10, 119, 142, 214, 442, 1145, 2092, 3126, etc.

• PC(X2) is the diagonal hyperbola with center X(664) passing through X(i) for these i : 2, 145, 174, 175, 176, 188, 508, 3152, 3177, 3210, etc. It obviously contains the vertices of the anticevian triangle of any of its points.

• PC(X9) is the circum-conic with perspector X(3900) passing through X(i) for these i : 2, 9, 200, 281, 282, 346, 2184, 2287, 2297, etc.