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X(4), X(13), X(14), X(30), X(1316)

Brocard points Ω1 and Ω2

other points below

O(X1316) is the Brocard (seventh) cubic. It is the only orthopivotal cubic passing through the Brocard points. See the FG paper "Orthocorrespondence and orthopivotal cubics" in the Downloads page and Orthopivotal cubics in the glossary. Also K001, property 3.

Its singular focus is X(98) and its asymptote is the line X(99)X(842), homothetic of the Euler line (which is the orthic line of the cubic) under h(X98, 2). The tangents at X(13) and X(14) concur at E on the Kiepert hyperbola.

K023a

K023 meets the circumcircle again at Y on the line X(110) X(1316).

Let A'B'C' be the circumcevian triangle of X(512). The parallels to the Euler line at A', B', C' meet the sidelines of ABC at U, V, W on the cubic.

The third intersection with the Fermat line is F3 on the Brocard circle.