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K592

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X(30), X(476), X(3233)

other points defined below

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. See other details at K593.

All these strophoids share the same real asymptote, namely the line X(30)X(125). Their singular foci F lie on the tangent at the vertex X(3233) of the Kiepert parabola.

The nodal tangents at P are the tangents drawn through P to the Kiepert parabola.

When P is the intersection of the Euler line and its perpendicular at X(110), the strophoid is right.

The inversive image of a strophoid in the circle with center P passing through F is a rectangular hyperbola.

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K592 is the locus of the real centers of anallagmaty denoted Ri, Si on the figure.

K592 is an axial nodal cubic symmetric in the line X(110)X(523) and with node X(476). The nodal tangents are the images of the line X(110)X(523) under the rotations around X(476) with angles +/– π/3.

It has three concurring asymptotes at X, the reflection of the vertex X(3233) of the Kiepert parabola about the Euler line. One of them is parallel to the Euler line.

K592 contains the traces A1, B1, C1 of the line X(3)X(125) on the sidelines of ABC.

The remaining points A2, A3 on BC lie on the circle passing through X(110) and X(476) whose center is the trace of the Euler line on BC. The points on CA and AB are defined likewise.

Naturally, the reflections of these nine points about the axis are nine other points on the curve. See figures below.

K592a K592b