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K593

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X(30), X(523)

The pedal curve of a point P on the Euler line with respect to the Kiepert parabola is a strophoid with node P. See K038 and K591 for instance. These curves are called Kiepert strophoids.

K593 is the locus of the real inflexion point.

K593 is an axial cubic symmetric about the line X(110)X(523). The three real (infinite) inflexion points are X(30) and X(523) which is double. Thus, K593 has three real inflexional asymptotes namely the line X(30)X(125) and the line X(110)X(523) counted twice.

The figure above shows two such strophoids with node Pi, singular focus Fi (midpoint of Pi-X110), meeting the common real asymptote X(30)X(125) on the perpendicular at Pi to PiFi. Each strophoid has a real contact Ti with the Kiepert parabola. The inflexion point is the intersection of the tangent at Ti to the parabola and the perpendicular at X(110) to Pi-X110.