Ga = A-vertex of antimedial triangle
A2 = A-vertex of the second Brocard triangle
infinite point of the median AG
Let P be a point. The triangles PAB and PCA have equal Brocard angle (and the same orientation) if and only if P lies on Ea (figure 1).
See below when the triangles PAB and PCA have opposite orientations.
Ea = K083-A is called A-equi-brocardian cubic and Eb, Ec 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 (which is the orthic line of Ea) at the intersection of GK and AX(76). The tangent at A is the symmedian AK.
Ea is an isogonal nK in the triangle AGaA2.
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, the anticomplement of the Kiepert hyperbola. (figure 2)
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) is the degenerate cubic above and E(K) is K018. See also a second group of equi-brocardian focals in K899.
All these cubics E(Q) pass through X(368) = equi-Brocard center (see TCCT p.267). See Table 61 for further properties and other cubics passing through X(368).
For any point Q on K018, E(Q) is a focal cubic whose singular focus F is also a point also on K018, namely the third point of K018 on the line KQ.
More generally, the singular focus F of E(Q) is the singular focus of the orthopivotal cubic O(Q). Recall that F is the reflection in an axis of the Steiner ellipse of the inverse of Q in the circle with diameter the foci of the inscribed Steiner ellipse. See Orthocorrespondence and Orthopivotal Cubics, §5. F can also be seen as the center of the polar conic of Q in the McCay cubic or in the Kjp cubic.
The focals Fa, Fb, Fc and three other equi-Brocard points
(with the collaboration of Jean-Pierre Ehrmann)
Let P be a point. The triangles PAB and PCA have equal Brocard angle (and opposite orientations) if and only if P lies on the focal cubic Fa. Fb and Fc are defined similarly.
The focus of Fa is A and its polar conic is the circle passing through A whose center is the antipode Oa of A on (O). The real asymptote is the sideline BC meeting Fa at the center Ωa of the A-Apollonius circle which is the tangential of A. The orthic line passes through the midpoints Mb, Mc of AC, AB and meets the polar conic of A at a1, a2 also on Fa.
It follows that Fa is an isogonal nK in the triangle Aa1a2, the locus of foci of conics inscribed in Aa1a2 whose center lies on the orthic line. Hence, the tangents at a1, a2 to Fa pass through A.
Ea, Fb, Fc generate a net of focal cubics having 5 common points (apart the circular points at infinity) hence they must meet at one real point A368 which is another equi-Brocard point. Similarly with the nets Eb, Fa, Fc and Ec, Fa, Fb we obtain B368 and C368. See figure below.
The net Ea, Fb, Fc contains a decomposed cubic which is the union of the line at infinity and the rectangular hyperbola Ha.
Ha contains :
• obviouly A368,
• B' : reflection of B in the perpendicular bisector of AC,
• C' : reflection of C in the perpendicular bisector of AB,
• Ai, Ae : intersections of (O) and the bisectors at A in ABC,
• the infinite points of these bisectors,
The center of Ha is the midpoint of B'C'.
Two other analogous rectangular hyperbolas Hb, Hc are defined passing through B368, C368 respectively.