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K680

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X(3), X(8), X(46), X(58), X(65), X(79), X(84), X(191), X(1782)

isogonal conjugates of X(2218), X(2915)

isotomic conjugate of X(2997)

vertices of the Fuhrmann triangle

Let A1B1C1 be the circumcevian triangle of the incenter X(1). The Fuhrmann triangle A2B2C2 is the triangle whose vertices are the reflections of A1, B1, C1 in the sidelines of ABC.

The cevian (resp. anticevian) triangle of P is perspective to the Fuhrmann triangle if and only if P lies on a circum-cubic Kc (resp. Ka) and, in both cases, the perspector lies on the Fuhrmann perspector cubic K680.

Kc passes through X(2), X(10), X(75), X(1737), X(2166) and Ka passes through X(1), X(2), X(6), X(106), X(1465), X(1718), X(2006). These two cubics have no notable properties.

K680 meets the line at infinity at the same points as pK(X6, X72) and these two cubics meet at six finite other points lying on the circum-conic with perspector X(2605).

K680 meets the circumcircle at the same points as K434 = pK(X6, X1770). These two cubics meet at three other points, namely X(46), X(79), X(191), on the trilinear polar of the isogonal conjugate of X(2605).