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X(98), X(511), X(10221), X(13414), X(13415) Brocard points Ω1 and Ω2 foci of the Kellipse (inellipse with center K when the triangle ABC is acutangle) foci of the inconic with center O, perspector X(69) more points and other figures below 

K019 = nK0(X6, X647) is the circular isogonal focal nK0 with root X(647) and singular focus the Tarry point X(98). See the related cubics K048, K417, K418. K019 is also spK(X511, X6) as in CL055. See K932, an analogous cubic. K019 is the locus of :




Points on K019 K019 contains : • A, B, C. • the intersections of the sidelines of ABC and the trilinear polar of the root X(647). • X(511), the infinite point of the Brocard axis OK. The real asymptote is the parallel at the Steiner point X(99) to the Brocard axis. • two other points on this line OK which are imaginary and also lie on pK(X6, X511). • the circular points at infinity. • the Brocard points Ω1 and Ω2. • the orthogonal projection E1 of X(98) on the line through Ω1 and Ω2. • X(98), the singular focus. • X = isogonal conjugate of E1, on the tangent at X(98), on the real asymptote, and on the circle through X(98), Ω1 and Ω2. X is the common tangential of X(98), X(511) and the Brocard points. • the four foci of the "Kellipse", the inconic with center K. • the four foci of the inconic with center O. • three points on the Euler line which also lie on Q002, the EulerMorley quartic, one of them E2 = X(10221) is always real. • three points on the Jerabek hyperbola (apart A, B, C) which also lie on Q003, the EulerMorley quintic, one of them E3 is always real. E3 is the isogonal conjugate of E2 = X(10221). • the six intersections of the altitudes and the cevian lines of O (Wilson Stothers). • two points S1, S2 on the line X(2)X(98) which are now X(13414), X(13415) in ETC (20170522). See also Table 62. These points S1, S2 are always real and lie on
• their isogonal conjugates S1*, S2* which lie on
• T1 = X(98)Ω1 /\ X(511)Ω2 and its isogonal conjugate T2 = X(98)Ω2 /\ X(511)Ω1. 



Hessian and prehessians of K019 

The hessian (H) of K019 contains O, H, X(648), the intersection of the Euler line with the trilinear polar of the root X(647). Note that the polar conic of X(648) in K019 is the union of these two latter lines. K019 has always three real prehessians (H1), (H2), (H3) and one of them – say (H1) – is a stelloid with radial center X(98), the singular focus of K019. The cubics – apart (H1) – share the same orthic line, namely the Brocard axis. Each prehessian is associated with a natural conjugation Fi on K019 which swaps a point and the center of its polar conic in (Hi). 



Conjugations on K019 

K019 is the isogonal pK with pivot X(511) with respect to the triangle Ω1 Ω2 X(98). Hence, K019 contains the in/excenters of this latter triangle which are four centers of anallagmaty on the cubic. These points lie on the polar conic of X(511), a rectangular hyperbola with center X(2968), passing through X(39) and X(512). It follows that K019 is now invariant under ten conjugations namely : • isogonal conjugation in ABC, • isogonal conjugation in Ω1 Ω2 X(98), • transformation JS (see Table 62) which maps a point M to the intersections of the polar lines of M in the Jerabek and Stammler rectangular hyperbolas, • three involutions F1, F2, F3 above, • four inversions with center one of the in/excenters of triangle Ω1 Ω2 X(98) that swap another in/excenter and the corresponding vertex of triangle Ω1 Ω2 X(98). 


