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X(1), X(3), X(40), X(2043), X(2044), X(11472), X(35237), X(45633)

X(39162), X(39163), X(39164), X(39165) : foci of the Steiner inellipse and their reflections in X(3)

excenters and their reflections in X(3)

infinite points of K243

other points below

Geometric properties :

K1293 is a member of the pencil containing K002, K800, K911.

K1293 and K002 meet at the in/excenters, the foci of the Steiner inellipse and O. These are the base-points of the pencil.

K1293 is a central cubic with center O and inflexional tangent passing through {3, 49, 155, 184, 185, 283, 394, 1092, 1147, 1181, 1204, 1216, etc}. See also K706, a very similar central cubic.

K1293 and K243 share the same points at infinity and meet again at six points on the Stammler hyperbola, namely the in/excenters, O and X(35237). These form another pencil that contains K609.

K1293 and (O) meet at six points, two by two symmetric about O, which lie on the parallel at O to the asymptotes of pK(X6, X15072).

Points on the bisectors of ABC

The remaining points of K1293 on the internal (resp. external) bisectors are the vertices of two triangles T1 = A1B1C1 (resp. T2 = A2B2C2) given by :

A1 = a^4-2 a^2 b^2+b^4-6 a^2 b c+4 b^3 c-2 a^2 c^2+6 b^2 c^2+4 b c^3+c^4 : 2 b^2 c (b+c) : 2 b c^2 (b+c),

A2 = a^4-2 a^2 b^2+b^4+6 a^2 b c-4 b^3 c-2 a^2 c^2+6 b^2 c^2-4 b c^3+c^4 : -2 b^2 c (b-c) : 2 b c^2 (b-c).

T1, T2 are both orthologic to the pedal triangle of X(376), with one common center of orthology X(11472). The other centers are not on the curve.

Points on the Brocard axis

With S = 2 area(ABC) and ω = Brocard angle,

Br1 = a^2 (a^2 b^2-b^4+a^2 c^2-c^4) - 2a^2 SA S Sqrt[3+cot²ω] : : , SEARCH = 11.8050583731608

Br2 = a^2 (a^2 b^2-b^4+a^2 c^2-c^4) + 2a^2 SA S Sqrt[3+cot²ω] : : , SEARCH = 1.75966741523184

Their barycentric product is a^4 (a^4-4 a^2 b^2+3 b^4-4 a^2 c^2+4 b^2 c^2+3 c^4) : : , SEARCH = 15.9706139543951, on the lines {X3, X2393}, {X50, X184}, {X160, X206}.