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

Brocard points Ω1 and Ω2

other points below

K023 = 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. See also K001, property 3.

Its singular focus is X(98) and its asymptote is the line X(30)X(99), homothetic of the Euler line (which is the orthic line of the cubic) under h(X98, 2). The tangential X of X(30) is the barycentric quotient X(385) ÷ X(542), hence it is the sixth point of K023 on the circum-conic with perspector X(385) passing through the Brocard points and X(83), X(99), X(880).

The tangents at X(13) and X(14) concur at E, on the lines {2,476}, {98,1989} and on the Kiepert hyperbola. E is the barycentric quotient X(1989) ÷ X(542), hence it lies on the circum-conic with perspector X(1989).

K023 meets the circumcircle again at Y = X(43654) on the lines {30, 805}, {74, 804}, {98, 14270}, {99, 37991}, {110, 1316}, {186, 22456}, {237, 476}, {401, 10420}, {419, 1304}, {523, 2698}, {542, 9160}, etc.

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 = X(18332) on the Brocard circle and on the lines {2,18331}, {3,690}, {30,9144}, {74,12042}, {98,5663}, {110,1316}, etc.