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K461

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X(3), X(8), X(3146), X(5895)

vertices of the cevian triangle of X(253)

infinite points of the altitudes

vertices of the 2nd Conway triangle, see ETC X(9776)

Any inscribed conic with perspector P on the Steiner ellipse is a parabola touching the sidelines of ABC at the vertices of the cevian triangle T of P. When P traverses the Steiner ellipse, the locus of the orthocenter of T is the cubic K461. See also K462 and a generalization at K654.

K461 is a crunodal cubic with node X(3146) and nodal tangents parallel to the axes of the Steiner ellipse or equivalently to the asymptotes of the Kiepert hyperbola.

K461 has three real asymptotes parallel to the altitudes of ABC and concurring at X(382).

These asymptotes meet the cubic again on a line which is the satellite of the line at infinity. This line is the homothetic of the parallel at X(382) to the Fermat line under h(X3146, 1/3).

K461 meets the sidelines of ABC at the vertices U, V, W of the cevian triangle of X(253), the isotomic conjugate of the de Longchamps point X(20). The other intersections are not always real and are symmetric about the feet of the altitudes.

The tangent at O is also parallel to the Fermat line. The polar conic of O is a rectangular hyperbola passing through O, X(3146) with asymptotes parallel to those of the Kiepert hyperbola.

Note that if P and Q are two antipodes on the Steiner ellipse, the corresponding orthocenters are collinear with O.

K461a

Let M be a fixed point. The locus of point P such that the line MP is perpendicular to the trilinear polar of the isotomic conjugate of P is a rectangular hyperbola H(M) passing through X2, X3413, X3414, M hence homothetic to the Kiepert hyperbola.

H(M) is a bicevian conic if and only if M lies on K461 in which case the perspectors P1, P2 lie on the Steiner ellipse (S) and on the isotomic transform (W) of the Wallace diagonal rectangular hyperbola respectively.

When isotomic is remplaced with isogonal, we obtain the analogous cubic K742.

***

More generally, if Ω is the pole of an isoconjugation lying on the Thomson cubic K002, the properties above obtained with Ω = X(2) for K461 and Ω = X(6) for K742 can be generalized as follows.

This study is inspired by a personal communication from Francisco Garcia Capitan.

If Ω and M are two fixed points, the locus H(M) of point P such that the line MP is perpendicular to the trilinear polar of the Ω-isoconjugate P* of P is a rectangular hyperbola passing through Ω, M and the infinite points of the rectangular circum-hyperbola passing through Ω.

• the tangent at M to H(M) is perpendicular to the trilinear polar of M*.

• if Ω = X4, H(M) splits into the line at infinity and the line HM, excluded in the sequel.

• if M = X4, H(M) is the rectangular circum-hyperbola passing through Ω, also excluded in the sequel.

Hence, for a given pole Ω, H(M) is a net of homothetic rectangular hyperbolas passing through Ω which contains :

• the rectangular circum-hyperbola H(Ω) passing through Ω,

• the degenerate rectangular hyperbola which is the union of the parallels at Ω to the asymptotes of H(Ω),

• a diagonal rectangular hyperbola if and only if Ω lies on K002,

• a rectangular hyperbola passing through the vertices of the orthic triangle if and only if Ω lies on K350 = K002-orthic.

***

H(M) is a bicevian conic if and only if M lies on a cubic K(Ω) generally not very interesting unless Ω lies on K002 in which case :

• K(Ω) passes through the vertices of the cevian triangle of a point X on the Lucas cubic K007. X is the X(4)-crossconjugate of taΩ, the isotomic conjugate of the anticomplement of Ω i.e. the perspector of the inconic with center Ω.

• K(Ω) is a nodal cubic with node N = X / (X2 / Ω) where / denotes Ceva conjugation.

• K(Ω) has three real asymptotes which are parallel to the cevian lines of X(4) in the cevian triangle of X(4) ÷ Ω (barycentric quotient).

With Ω = X(2), we have X = X(253) and N = X(3146) as mentioned above, H(Ω) being the Kiepert hyperbola.

With Ω = X(6), we have X = X(4) and N = X(155) as mentioned in the page K742, H(Ω) being the Jerabek hyperbola.