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The equi-Brocard center X(368) is the center P for which the triangle PBC, PCA, PAB have equal Brocard angles (see TCCT p.267).

The triangles PCA and PAB have equal Brocard angles (and the same orientation) if and only if P lies on Ea = K083-A. Eb = K083-B, Ec = K083-C are defined similarly.

Ea is a focal cubic with singular focus A2, vertex of the second Brocard triangle. The real asymptote is parallel to the median AG at the intersection of GK and AX(76). The tangent at A is the symmedian AK.

Ea, Eb, Ec generate a net of circular cubics containing the Brocard (second) cubic K018 and another degenerate cubic into the union of the line at infinity and the Wallace hyperbola (W), the anticomplement of the Kiepert hyperbola. All these cubics pass through X(368).

Each cubic E(Q) of the net can be written under the form : E(Q) = p Ea + q Eb + r Ec, where Q = p : q : r is any point.

In particular, E(G) = E(X2) is the degenerate cubic above and E(K) = E(X6) is K018.

The singular focus F of E(Q) is the singular focus of the orthopivotal cubic O(Q) i.e. F = Psi(Q). Recall that Psi(Q) is the reflection in the focal axis of the Steiner ellipse of the inverse of Q in the circle with diameter the real foci of the inscribed Steiner ellipse. Psi(Q) can also be seen as the center of the polar conic of Q in the McCay cubic or in the Kjp cubic. See here for a list of pairs {Q, F}.

When Q = X(6), E(Q) = O(Q) = K018.

For Q ≠ X(6), E(Q) and O(Q) generate a pencil of circular cubics which contains the decomposed cubic which is the union of the line at infinity (twice) and the line X(6)Q.

For any point Q ≠ X(6) on K018, E(Q) and O(Q) are both focal cubics whose common singular focus F is a point also on K018, namely the third point of K018 on the line X(6)Q.

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Properties of E(Q)

• E(Q) contains Q, X(368), the circular points at infinity J1, J2 with tangents passing through the singular focus F = Psi(Q).

• The third (real) point J at infinity – assuming Q ≠ X(2) – is that of the line X(2)Q which is the orthic line (L) of the cubic.

• E(Q) must meet (L) at a third real point Q' which lies on the Wallace hyperbola (W).

• The real asymptote (A) of E(Q) is then the image of (L) under the homothety h(F, 2).

• When Q ≠ X(6), E(Q) and O(Q) are distinct and meet at three (double) points at infinity hence at three other finite points which are Q and two other points M, N on the line X(6)Q. These two points also lie on a circum-conic (C) passing through X(2).

The following table (built with the collaboration of Peter Moses) shows a selection of the most remarkable cubics E(Q) with focus F. O(Q) is also mentioned and X(368) is not repeated in the lists of centers.

Remarks :

• for any Q on the Brocard axis, E(Q) passes through X(15), X(16) and F lies on the Parry circle. See pink cells.

• for any Q on the Fermat axis, E(Q) passes through X(13), X(14) and F lies on the circle passing through X(2), X(13), X(14), X(111), X(476) with center X(8371). See light blue cells.

• for any Q on the Euler line, E(Q) passes through X(20), X(30) and F lies on the line X(2), X(98).

• for any Q on the line X(2), X(98), E(Q) passes through X(147), X(542) and F lies on the Euler line.

• for any Q on the line X(6), X(110), E(Q) passes through X(6), X(111) and F lies on the circle passing through X(2), X(3), X(6), X(111), X(691) with center X(9175). See light green cells.

Recall that E(Q) is a focal cubic when Q lies on K018 in which case F also lies on K018.

Note that most of the listed cubics belong to one of the three pink, blue, green families and that K018 belongs to all of them.