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A cubic (K) = cK(#X1, P) is an isogonal nodal conicopivotal cubic. See Special Isocubics ยง8 for general properties. For any M on (K), its isogonal conjugate M* also lies on (K) and the line MM* (when defined) is tangent to a conic called the pivotal conic (PC) of (K). Here, (PC) is inscribed in the excentral triangle for any P. (K) meets the sidelines of ABC again at U, V, W on the trilinear polar (L) of the root P and then (PC) is also inscribed in the triangle bounded by the lines AU, BV, CW. M and M* are then conjugated with respect to a fixed circle (C) passing through X(1) which is orthogonal to the circles with diameters AU, BV, CW. X(1) is a node on (K) and the nodal tangents are the tangents drawn from X(1) to (PC). Let (I) be the circumconic with perspector X(1) and center X(9), passing through X(88), X(100), X(162), X(190), X(651), X(653), X(655), X(658), X(660), X(662), X(673), X(771), X(799), X(823), X(897), X(1156), X(1492), X(1821), X(2349), X(2580), X(2581), etc. (K) is acnodal (resp. crunodal) when P is interior (resp. exterior) to (I). When P lies on (I), (K) splits into a line and a circumconic. *** Special cubics (K) โข (K) is equilateral if and only if P = X(4383) and then it is K085. โข (K) is a K0 (no term in x y z) if and only if P lies on the antiorthic axis, the trilinear polar of X(1). In this case, it is the X(1)Hirst inverse of the circumconic with perspector P and also the X(1)line conjugate of the bicevian conic C(X1, P*). โข (K) is circular if and only if P lies on the trilinear polar of X(7). In this case, the nodal tangents are perpendicular and the singular focus F lies on (K) and on the circumcircle (O). Hence (K) is a strophoid. (PC) is a parabola whose focus is the reflection of X(1) in F and whose directrix is the orthic line of (K) i.e. the parallel at X(1) to the real asymptote of (K). See CL003 for further geometric properties. *** A construcion of (K) (K) is the image of the circumconic (I) under the transformation ๐_{P} which is described in page K1065. With P = u : v : w, this transformation is given by ๐_{P} : X = x : y : z โ X' = x^2 (w y  v z) : y^2 (u z  w x) : z^2 (v x  u y).




The following table gives a selection of these cubics cK(#X1, P). The red point is the singular focus of a strophoid. 



note 1 : K228 is a circumconico pivotal cubic. (PC) is the circumellipse (I) with perspector X(1) mentioned above. note 2 : the nodal tangents are parallel to the asymptotes of the Kiepert hyperbola. note 3 : the nodal tangents are parallel to the asymptotes of the Jerabek hyperbola. note 4 : the nodal tangents are parallel to the asymptotes of the Feuerbach hyperbola.

