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X(1), X(4), X(36), X(80), X(517), X(953)

K681 is a member of the pencil of circular circum-cubics passing through X(1), X(4), X(80), X(517) which we call Fuhrmann-Feuerbach cubics. This pencil is generated by K334 and the union of the line at infinity with the Feuerbach hyperbola. See also K529, K682, K683, K684, K685.

K681 is a focal cubic with focus the incenter X(1). The tangent at X(1) passes through the orthocenter H.