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These points Ua, Ub, Uc are defined in Special Isocubics §6.5. They are the only points for which there are infinitely many equilateral pKs with pivot one of these points. They lie on a large number of curves although they are not constructible with ruler and compass only. See also CL052 and Table 41. 

Ua, Ub, Uc are the reflections of the orthocenter H about the points Na, Nb, Nc which are the intersections (apart A, B, C) of the Napoleon cubic K005 and the circumcircle of ABC. In other words, the triangle UaUbUc is the homothetic of NaNbNc under the homothety with center H and ratio 2. *** Let M be a point different of O and let M* be its isogonal conjugate. The vectors MM* and OH are equal if and only if M is one of the points Ua, Ub, Uc. *** These points Ua, Ub, Uc and their isogonal conjugates Ua*, Ub*, Uc* lie on the Neuberg cubic K001 and also on K732 which are two isogonal cubics. See also the associated points Va, Vb, Vc and Va*, Vb*, Vc* on the Neuberg cubic. At the very bottom of this page, a table gathers all the related cubics together. 



Conics Three hyperbolas 

Ga, Gb, Gc are the vertices of the antimedial triangle. Denote by (Ha) the hyperbola passing through B, C, Ga, the reflection A' of A in the line BC, the reflection Ah of H with respect to the second intersection of the altitude AH with the circumcircle. The asymptotes of (Ha) make 60° angles with the sideline BC hence (Ha) is a hyperbola with eccentricity 2. The center Oa is the midpoint of GaAh, a point on the perpendicular bisector of BC. (Hb) and (Hc) are defined similarly. These hyperbolas generate a net of conics passing through three points Ua, Ub, Uc. 

Other conics of the net Every conic of the net has the form : p (Ha) + q (Hb) + r (Hc) where P = (p:q:r) is any point in the plane. Obviously, P = A, B, C give the three hyperbolas. The net contains : • only one circle with P = X(69). This is the circle C(L,2R) where L = X(20) is the de Longchamps point. • a pencil of rectangular hyperbolas obtained when P lies on the de Longchamps line (trilinear polar of X(76), isotomic transform of the circumcircle). These rectangular hyperbolas always pass through O and are centered on the circle C(X(550), R) whereX(550) is the reflection of X(5) in O. The most remarkable are obtained when :
• a family of parabolas obtained when P lies on an ellipse centered at the intersection of the lines GK and X(30)X(76). This ellipse does not contain any ETC center up to X(3053). When P is inside, outside this ellipse, the conic of the net is an ellipse, a hyperbola respectively. • a family of decomposed conics when P lies on the reunion of three lines forming a triangle circumscribing the ellipse above. 



Cubics Circumcubics obtained with cevian lines Let P = p:q:r be a point and (La), (Lb), (Lc) its cevian lines. Any cubic of the form (La) (Ha) + (Lb) (Hb) + (Lc) (Hc) is a circumcubic passing through P and the points Ua, Ub, Uc. All these cubics form a net which contains :
*** Circumcubics obtained with anticevian lines Let P = p:q:r be a point and (La), (Lb), (Lc) the sidelines of the anticevian triangle of P. Any cubic of the form (La) (Ha) + (Lb) (Hb) + (Lc) (Hc) is a circumcubic passing through and the points Ua, Ub, Uc. All these cubics form a net which contains :
*** Equilateral circumcubics circumscribing the antimedial triangle K092 and K142 generate a pencil of equilateral circumcubics passing through the points Ua, Ub, Uc and the vertices Ga, Gb, Gc of the antimedial triangle. This pencil contains K080 = KO++, the only K+. The equilateral triangle formed by the asymptotes has always center O. *** Circular circumcubics through O K330 and K331 generate a pencil of circular circumcubics passing through the points Ua, Ub, Uc and O. This pencil contains :


Figure 1 : nK cubics nKa has singular focus the reflection Ao of A in O. The reflection of O in Ao lies on the curve. nKa is tangent at A to the circumcircle. It meets the sidelines AB, AC at two points lying on a parallel to BC. Its pole lies on AK and its root on AG. 

Figure 2 : focal cubics All three cubics Fa, Fb, Fc pass through O. Fa has singular focus the point A. The reflection Ao of A in O lies on the curve. Fa contains the midpoint of BC and Ga, the Avertex of the antimedial triangle. Its real asymptote is the perpendicular to BC at X(1657), the reflection of O in L. 

*** Cubics passing through O and H K001 and K080 generate a pencil of circumcubics passing through the points Ua, Ub, Uc, O and H. This pencil contains the Lemoine cubic K009. See table 16 : the NeubergLemoine pencil. 

Summary of points on cubic curves 



Two circumcubics in cells of the same color are isogonal transformed from each other. K001 and K732 are selfisogonal cubics. 



Other curves These points Ua, Ub, Uc lie on : Q050 a circular quintic through G, O, H and the Fermat points Q051 a circular quartic through G and the Fermat points Q052 a circular quartic through G, O

