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K460

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X(2), X(524), X(623), X(624), X(1352), X(3413), X(3414)

X(3413), X(3414) : infinite points of the Kiepert hyperbola

complements of X(1513), X(3094)

other points below

For any point M on the Kiepert hyperbola (K), the isotomic conjugate M' of M lies on the line GK, the tangent at G to (K). When M traverses (K), the midpoint of MM' lies on the Kiepert cuspidal cubic K460.

K460 has a cusp at G with cuspidal tangent passing through X(99), X(111), etc. It has only one inflexion point namely X(524), the infinite point of the line GK.

K460 is invariant in the oblique symmetry with axis the cuspidal tangent and direction GK.

K460 has three real asymptotes : two are the parallels at X(620) to those of (K) and one is the parallel at X(115) to the line GK. X(620) is the complement of the center X(115) of (K). Note that the cusp G must lie on the Steiner inellipse of the triangle formed by the asymptotes since its polar conic is a decomposed parabola. Recall that this ellipse is the poloconic of the line at infinity in K460.

Compare K459 and K460.

K460 is the image of the complement of the Kiepert hyperbola under the involution KW described in the page Table 62.

K460a

Let M, N be the points with abscissa 4/3, -1/3 in (G, X115) respectively.

The parallels at X(148) to the asymptotes passing through X(3413), X(3414) meet K460 at M1, M2 on the parallel at M to the third asymptote, that passing through X(524).

The parallel at N to the asymptote X(115)X(524) is the satellite line of the line at infinity. It meets the two other asymptotes at N1, N2.

M, N are the midpoints of M1, M2 and N1, N2 respectively.

Generalization

Let P be a fixed point on (K). The isoconjugation with fixed point P maps (K) to its tangent (T) at P. For any point M on (K), its isoconjugate M* lies on (T) and the midpoint of MM* lies on a cuspidal cubic with cusp P. This cubic has three real asymptotes : two are parallel to those of (K) and one is parallel to (T).