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X(2), X(6), X(13), X(14), X(15), X(16), X(30), X(74), X(378), X(3580), X(4846), X(14910)

E1, E2 on the circumcircle and on the line X(30)-X(115)

E1*, E2* their isogonal conjugates at infinity and on the circum-conic with perspector X(34291)

K505 is a remarkable isogonal nK cubic whose root X(14611) lies on the lines X30-X146, X110-X476, X147-X858, X648-X1302, etc. It is a member of the class CL061.

The polar conic of X(74) contains X(13), X(14), X(15), X(16), X(74) and X(2132). In other words, the tangents to K505 at the Fermat and isodynamic points concur at X(74), the isogonal conjugate of the infinite point of the Euler line. It follows that K505 is a pK with pivot X(74) invariant under the isoconjugation with respect to the (improper) triangle X(2)X(6)X(30). Note that X(2)X(6)X(30) is the diagonal triangle of X(13), X(14), X(15), X(16). See also the related K881 and K1191.

The polar conic of X(14910) contains X(2), X(6), X(30), X(74) and the diagonal triangle of these four points is X(378)X(3580)X(4846). Hence K505 is also a pK with pivot X(14910) in this triangle.

K505 is the locus of point P whose pedal circle is orthogonal to the fixed (orange) circle with center at the intersection of the Simson lines of X(74), E1, E2. See Special Isocubics § 1.5.5.

K505 shares nine common points with the Neuberg cubic K001 namely A, B, C, X(13), X(14), X(15), X(16), X(30), X(74). The pencil P generated by K001 and K505 is invariant under isogonal conjugation : the isogonal transform of any member is another member, K001 and K505 each being its own transform.

P contains a pair of (always real) nKs and a pair of psKs, each pair is invariant under isogonal conjugation.

Any cubic of P that meets the circumcircle (O) at P1 (different of X74) meets (O) again at P2 such that P1, P2, X30 are collinear.

K802 and its isogonal transform K803 are two other remarkable members of P.


K505 is also a member of the pencil nK(X6, S, X2) where S is a point on the trilinear polar of the barycentric product X99 x X476, a line passing through X(110), X(476), X(523), X(3233), X(6742), X(7471), etc.

• nK(X6, X110, X2) is the union of the Brocard axis and the Kiepert hyperbola,

• nK(X6, X476, X2) is the union of the Fermat axis and the circum-conic with perspector X(526) passing through X(2), X(15), X(16), X(186), X(249), X(323), X(842), X(1138), X(2411), etc, the isogonal transform of the Fermat axis.

• nK(X6, X523, X2) is the only nK0 namely K018.

• nK(X6, R, X2) with R = X(14985) on the lines X(11)X(2607), X(100)X(190), etc, is a cK with node X(1).