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K1241

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X(3), X(5), X(182), X(206), X(5092), X(44882), X(44883), X(44884), X(44885)

infinite points of the altitudes

vertices G1, G2, G3 of the Grebe triangle

foci of the inconic with center X(3589), perspector X(83)

X(44882), X(44883) = reflections of X(5), X(206) in X(5092)

other points below

Geometric properties :

K1241 is the image of K644 under the isogonal conjugation with respect to the Grebe triangle.

It is a central cubic with center X(5092) having three real asymptotes passing through X(5092) and perpendicular to the sidelines of ABC. X(5092) is X(140) of G1G2G3.

K1241 passes through the vertices of the Grebe triangle and the tangents at these points concur at X =a^2 (2 a^6 b^2-2 a^2 b^6+2 a^6 c^2-5 a^2 b^4 c^2-b^6 c^2-5 a^2 b^2 c^4-6 b^4 c^4-2 a^2 c^6-b^2 c^6) : : , SEARCH = 6.30927110640310, a point on the Euler line but not on the cubic. It follows that K1241 is a psK in the Grebe triangle hence it must meet its sidelines again at three points R1, R2, R3 which are the vertices of a cevian triangle of G1G2G3, that of X(5085). Note that R1R2R3 is the pedal triangle of O in G1G2G3. X(5085) is the centroid of G1G2G3.

K1241 meets the circumcircle (O) again at S1, S2, S3 which are the antipodes of Q1, Q2, Q3 on (O). These latter points lie on K836 (the isogonal transform of K644), on K655, and, in fact, on infinitely many pK(Ω, P) with Ω on pK1 = pK(X32 x X3108, X3108) and P on pK2 = pK(X3108, X76 x X3108). Note that pK1 is the barycentric product X(6) x pK2.

S1, S2, S3 lie on nK0(X6, R) with R = SB SC (a^2 - 2SA) + 2a^2 SO^2 : : , SEARCH = 1.02268009044742, a point on the line {6,468}.