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X(1), X(4), X(46), X(80), X(119), X(517), X(915), X(1785), X(1845)

feet of the altitudes

K334 is a pK invariant under orthoassociation i.e. inversion in the polar circle. It is a circular cubic, a member of the class CL019.

Its singular focus is the Feuerbach point X(11) whose antipode on the nine-point circle X(119) lies on the cubic. Its real asymptote is the line X(119)-X(517).

The polar conic of the infinite point X(517) is the rectangular circum-hyperbola which is the isogonal transform of the line OX(8). Its center is the second intersection of the asymptote and the nine-point circle. The tangents at A, B, C, H to the cubic are all parallel to the line OI.

The isogonal transform of K334 is K436, an inversible cubic.

K334 and the union of the line at infinity with the Feuerbach hyperbola generate a pencil of circular circum-cubics passing through X(1), X(4), X(80), X(517) which we call Fuhrmann-Feuerbach cubics. Each cubic has its singular focus on the line X(1)X(5) and its orthic line is the line X(5)X(10).

This pencil also contains the two focal cubics K529, K681 with foci X(80), X(1) respectively, the cubic K682 with focus X(5) and concurring asymptotes. See also K683, K684, K685.