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K591

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X(4), X(30), X(113)

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 with 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 with isogonal cK cubics.