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X(1), X(40), X(84), X(164), X(505), X(2956), X(3645),X(6212), X(6213)

excenters

K414 = pK(X6, X40) is the Orthocubic for the excentral triangle IaIbIc. Hence, it is the locus of M such that :

– I (the incenter), M and the isogonal conjugate of M with respect to the excentral triangle are collinear,

– X(40) (the circumcenter of the excentral triangle), M and the isogonal conjugate of M are collinear.

Hence K414 is an isogonal pK in two different triangles like K006 and K415. See also K878 for a generalization and other related cubics.

K414 meets the Orthocubic at A, B, C, the in/excenters and X(6212), X(6213), two isogonal points on the line HX(40) and on the circumconic through O and X(84).