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X(4), X(30), X(113), X(5011), X(5167), X(15341), X(15427), X(31862), X(31863), X(43395), X(43396), X(45272), X(52219), X(56884)

vertices of the orthic triangle

K591 is the pedal curve of the orthocenter H with respect to the Kiepert parabola. Compare with K038 where H is replaced by O. These two curves are two examples of Kiepert strophoids. More informations at K592 and K593.

K591 is also related to the Neuberg cubic as follows : a line passing through H meets the Neuberg cubic again at two points M, N and their midpoint P lies on K591.

K591 is a strophoid with node H, singular focus X(113), asymptote parallel to the Euler line at X(125), the center of the Jerabek hyperbola.

The orthoassociate (inversive image in the polar circle) of K591 is the rectangular circum-hyperbola with center X(136), perspector X(2501) that contains X(93), X (225), X (254), X (264), X (393), X (847), X (1093), X (1105), X (1179), X (1217), X (1300), X (1826) giving as many points on K591. This hyperbola is tangent at H to the Euler line. Note that the orthoassociate of X(1300) is X(113).

When ABC is acute angle, K591 is the Gergonne strophoid K086 of the orthic triangle giving other properties related to isogonal cK cubics.

See the analogous strophoids K955, K1300. K1397 is the excentral version of K591.

A generalization

Let (H) be the rectangular circum-hyperbola with perspector P, on the orthic axis, and center Ω = G-Ceva conjugate of P, on the nine-point circle.

The orthoassociate of (H) is the strophoid (S) with node H which passes through the vertices Ha, Hb, Hc of the orthic triangle.

(H) meets (O) again at M, the trilinear pole of the line X(6)P and the reflection of H about Ω. The orthoassociate of M is the singular focus F of (S), a point on the nine-point circle.

The isogonal conjugate of the anticomplement of F is the real infinite point ∞S of (S).

The perpendicular at H to the line HΩ meets (H) again at Y and the orthoassociate X of Y is the point, apart ∞S, where (S) meets its real asymptote.

The line HΩ meets (O) at M and another point Z. (S) is the pedal curve of the orthocenter H with respect to the parabola (P) with focus Z and directrix the parallel at H to the real asymptote, namely the line H∞S. (P) is the parabola, inscribed in ABC, with perspector the barycentric quotient H÷P.

(S) meets the sidelines of HaHbHc at three collinear points Pa, Pb, Pc on the line (L) which is the trilinear polar with respect to HaHbHc of the barycentric product HxP. (S) is the isogonal cK with root HxP with respect to HaHbHc.

Remarks :

• The nodal tangents at H to (S) are tangent to (P).They are obviously parallel to the asymptotes of (H).

• The Simson line of Z is the reflection about H to the real asymptote. Hence the envelope of this real asymptote is the deltoid which is the reflection of the Steiner deltoid H3 about H.

• This Simson line of Z is also tangent to (P).