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circular points at infinity

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K899-A, K899-B, K899-C are the three focal cubics E(A2), E(B2), E(C2) in Table 61 where A2B2C2 is the second Brocard triangle. Their singular foci are A, B, C respectively. See the other equi-brocardian focals K083-A, K083-B, K083-C.

Properties of K899-A

• The singular focus is A with tangent passing through the trace on BC of L(X512), the trilinear polar of X(512).

• The polar conic (Ca) of A is the circle passing through A, A2 and A2', where A2' is the second point of the orthic line GA2 on the Wallace hyperbola.

• The real asymptote is the parallel at X(99) to the orthic line and passes through Ka, the A-vertex of the circumcevian triangle of X(6).

• K899-A contains Xa, intersection of the real asymptote and the tangent at A.

• K899-A contains Ya, intersection of the perpendicular bisector of A2A2' and the line AX(671) which is parallel to the asymptote.

• K899-A is an isogonal nK in the triangle AA2A2'.


K899-A, K899-B, K899-C generate a net of circular cubics passing through X(368) and each cubic of the net may be written under the form :

K899-P = p K899-A + q K899-B + r K899-C,

where P = p : q : r is any point.

With P = X(2), the cubic splits into the line at infinity and the Wallace hyperbola.

With P = X(111), the cubic is K018.