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∑ (b^2 + c^2) x^2 (c^2y - b^2 z) = 0

X(2), X(4), X(6), X(83), X(251), X(1176), X(1342), X(1343)

vertices G1, G2, G3 of the Grebe triangle

infinite points of K655 = pK(X6, X141)

A'B'C' cevian triangle of X(83)

F, F' foci of the in-conic with perspector X(83), center X(3589)

K644 is the locus of pivots of pKs passing through the vertices of the Grebe triangle. The locus of isopivots is K177 and the locus of the poles is K1258 = pK(X251 x X32, X251). See also Table 57.

K644 is another example of cubic passing through the vertices G1, G2, G3 of the Grebe triangle and here, the tangents are concurrent at X on the lines X(2)X(1285) and X(83)X(183) with first barycentric 5 a^4+7 a^2 b^2+7 a^2 c^2+8 b^2 c^2 and SEARCH = 2.07253149618850. X is now X(14535) in ETC (2017-09-25).

The tangents at A', B', C', K pass through G and the tangents at A, B, C, X(83) pass through K. The tangent at G is the Euler line.

K644 meets the in-conic with perspector X(83) at A', B', C' and three other points which are the contacts of this conic with the sidelines of the Grebe triangle. In other words, this conic is inscribed in ABC and in G1G2G3.

K644 is spK(X141, X3589) as in CL055. It is a psK with respect to the Grebe triangle. Indeed, the tangents at G1, G2, G3 concur at X(14535) but this point is not on the cubic.

K644 is the locus of P whose anticevian triangle is perspective (at Q) to the circumcevian triangle of X(2). Q lies on K959.


The isogonal transform of K644 is K836 = pK(X39, X2).

K644 is transformed into K1241 under the isogonal conjugation with respect to the Grebe triangle.

Note that the two isogonal conjugates of M on K644 lie on a line passing through O.