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X(4), X(20), X(175), X(176), X(1131), X(1132) the eight Soddy centers i.e. X(175), X(176) and associated points, see a figure below the reflections A', B', C' of A, B, C in BC, CA, AB the intersections of the lines through the midpoints of the altitudes and X(20) with the corresponding sidelines of ABC 

L = X(20) is the de Longchamps point of triangle ABC. For any point P, denote by La de Longchamps point of triangle PBC and define Lb, Lc similarly. Triangles ABC and LaLbLc are perspective if and only if P lies on the Soddy cubic (JeanPierre Ehrmann). K032 is a member of the pencil of cubics generated by the Neuberg cubic and the union of the altitudes. See "On two remarkable pencils of cubics" in the Downloads page. The asymptotes of K032 are parallel to those of K156, isogonal pK with pivot P = X(5059). Since this point P is always outside the deltoid which is the envelope of axes of inscribed parabolas, there is only one real asymptote and two other imaginary. The six other common points are A, B, C, X(4), X(1131), X(1132) on the Kiepert hyperbola. It follows that the pencil of cubics generated by K156 and the union of the line at infinity and the Kiepert hyperbola also contains K032. Recall that K156 is a member of the Euler pencil of cubics. See Table 27. K032 meets the circumcircle (O) at the same points as pK(X6, X12279). The three remaining common points lie on the Brocard axis. K032 and the Neuberg cubic share the same tangent (passing through X54), the same polar conic (a rectangular hyperbola passing through X4, X20, X393, X3164, homothetic to the rectangular circumhyperbola with perspector X2501), the same (complicated) osculating circle at H. K032 and the Neuberg cubic also share the same orthic line, namely the Euler line. Hence, the polar conic of X(20) in K032 is also a rectangular hyperbola passing through X(20), X(347), tangent at X(20) to the Euler line, homothetic to the rectangular circumhyperbola passing through X(3146). 



K032 and the CPCC points The Soddy cubic K032 is also a member of the pencil of cubics generated by the Darboux cubic and the Lucas cubic. See table 15. It contains the four CPCC points (blue points on the figure), see table11. These points lie on the rectangular hyperbola (H) passing through X(5), X(6), X(20), X(69), X(1498) whose asymptotes are parallel to those of the Jerabek hyperbola. The "last" common point with K032 has rather ugly coordinates. 



K032 and the Soddy points The Soddy centers are X(175) and X(176). Each Soddy center has three extraversions denoted Xa, Xb, Xc on the figure. These eight points are two by two collinear with the de Longchamps point X(20). 


