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K1092

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X(6), X(30), X(1379), X(1380), X(5000), X(5001), X(11472), X(13872), X(16303)

circular points at infinity

foci of Steiner inellipse

O5, O6 : isogonal of the MacBeath circumconic infinite points, on the orthic axis, (O), nine points circle, etc

points of inflexion of K006

Geometric properties :

See K1091, locus property 2. Recall that if Ω is the center of a circle (𝚪) of the pencil generated by (O) and the nine points circle then the line ΩX(6) meets (𝚪) at two points on K1092.

The Orthocubic K006 and its Hessian cubic (H) generate a pencil of cubics sharing the same nine points of inflexion. This pencil contains the focal cubic K1092, a stelloid (S) whose Hessian cubic is K1092 and two (always real) K+ with concurring asymptotes at X(6) which is the singular focus of K1092 and the radial center of the stelloid. See a first generalization in K627 and another one below.

Note that the asymptotes of the stelloid are the parallels at X(6) to those of the McCay cubic K003.

Note : O5, O6 are the bicentric pair P(4), U(4) in ETC and they are X(2)-Ceva conjugates. They also lie on the cubics K397, K489, K490, K491, K492, K493, K496, K533, K535, K816, K817, K818, K1091.

K1092a

When ABC is obtusangle (figure above), O5 and O6 are real , X(5000) and X(5001) are complex conjugates. K1092 is bipartite.

It has four real centers of anallagmaty on the rectangular hyperbola (C) and on the parallel at X(6) to the asymptotes of the Jerabek hyperbola. (C) is the polar conic of X(30) with center X(2453), passing through X(30), X(523), X(468).

 

When ABC is acutangle (figure opposite), O5 and O6 are complex conjugates, X(5000) and X(5001) are real. K1092 is unipartite.

 

The brown circle is the polar conic of the singular focus X(6). It contains X(6), X(111), X(112), X(115), X(187), X(1560), X(2079), X(5000), X(5001), etc. This is the Moses-Parry circle mentioned in ETC, article X(8428).

K1092b

K1092 is the isogonal pK with pivot X(30) with respect to the triangle (T) with vertices X(6), O5, O6. This triangle is a proper triangle when ABC is obtusangle.

The corresponding isogonal conjugates X(1379)*, X(1380)* of X(1379), X(1380) are the common points of the Fermat axis and the orthocentroidal circle (dashed blue curves). They also lie on the cubics K025, K492, K591, K800.

Note that X(1379)*, X(1380)* lie on the axes of the Steiner inellipse whose foci are four other points on K1092 which obviously also contains their isogonal conjugates in (T).

X(1379)*, X(1380)* are now X(31863), X(31862) in ETC (2019-03-27).

The line X(2)X(6) and the orthoptic circle of the Steiner inellipse meet at two other points on K1092 (dashed green curves).

 

Another generalization

The Orthocubic K006 = pK(X6, X4) may be replaced with any (K) = pK(X6, P) with P on the Darboux cubic K004 giving analogous properties.

Indeed, (K) and its Hessian cubic (H) generate a pencil of cubics which always contains a stelloid (S) and a focal cubic (F) whose singular focus F is the radial center of (S). All these cubics obviously share the same points of inflexion. The asymptotes of (S) are parallel to those of the McCay cubic K003.

F is the perspector of ABC and the medial triangle of the pedal triangle of P hence it is a point on the Thomson cubic K002 (see locus property 15 in K004).

Special case : with P = X(3) hence F = X(2), (K) is K003, which is also the stelloid (S), and (H) is K048 which is also the focal (F).