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too complicated to be written here. Click on the link to download a text file. 

traces of the trilinear polar (L) of X(323) vertices of the circumnormal triangle 

K404 is an example of CircumNormal nK0. See Table 25. K404 and K003 meet at 6 points on the circumcircle and then 3 other points on the line (L') passing through X6, X1511, X2079, X2931, etc. Following Poncelet, there must be a conic (C) inscribed in ABC and in the circumnormal triangle. (C) is the inconic with center X(140). It is tangent to the trilinear polars of X(323) and tgX924, the isotomic of the isogonal of the pole X(924). *** More generally, any nK0 with pole Ω at infinity and root X(323) meets the circumcircle again at the vertices of an equilateral triangle (T). The inconic (C) inscribed in (T) has its center on the line X(140)X(523) and is tangent to the trilinear polars of X(323) and tgΩ. Examples : • nK0(X511, X323) is a CircumTangential nK0 and (C) is the Kiepert parabola. • nK0(X526, X323) is the isogonal transform of the stelloid nK0(X110 x X1989, X110). 
