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K731

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X(6), X(83)

vertices of the Grebe triangle

foci of the inconic with center the midpoint of X(6)-X(39)

infinite points of K326 = pK(X6, X39)

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

K731 is a member of the pencil of cubics generated by the Grebe cubic K102 and the union of the circumcircle and the Brocard axis OK.

It contains the vertices G1, G2, G3 of the Grebe triangle and the tangents at these points concur at X(6). Recall that these tangents are the altitudes of the Grebe triangle since X(6) is its orthocenter. See also Table 57.

The polar conic (JG) of X(6) is the Jerabek hyperbola of the Grebe triangle. The tangent at X(83) also passes through X(6).

K731 is a pK with respect to G1G2G3 with pivot X(83) and isopivot X(6) hence K731 passes through the vertices of the cevian triangle of X(83) in G1G2G3.

K731 meets the sidelines of ABC at A', B', C' and A' also lies on the parallel at X(6) to the line AX(39).