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vertices of the first Brocard triangle

points at infinity of ABC sidelines (inflexions)

X = [ a^2(b^2 - c^2)^3 : : ] tangential of K

Regard a triangle ABC and a point P. If x1, y1, z1are the distances from P to the sidelines BC, CA, AB (and x1 > 0 for P on the same half-plane of BC as A, etc.), and k is a given constant, then the locus of all points P so that x1 y1 z1 = k is a cubic with trilinear equation (2D)^3 xyz = k (ax+by+cz)^3, where ( x : y : z ) are the trilinear coordinates of P, and D is the area of triangle ABC. (Darij Grinberg, Hyacinthos #8084).

All those cubics have the sidelines of ABC as inflexional asymptotes (Jean-Pierre Ehrmann) and form a pencil of cubics.

K153 is the only one passing through K and also the vertices A1, B1, C1 of the first Brocard triangle. It is the locus of point P such that the product of (signed) distances from P to the sidelines of ABC is abc/8 tan^3 w, where w is the Brocard angle. The polar conic of K is the circle centered at O, homothetic of the Brocard circle under h(K,2). The tangent at A1 meets B1C1 at A2 on the curve, B2 and C2 are defined similarly. The tangent at K is the trilinear polar of the Parry point X(111) and is the perpendicular at K to the Brocard line.