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K893

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X(2), X(3), X(6), X(110), X(111), X(542), X(2482)

E = Psi(X2482) = X(14916)

foci of the Steiner inellipse

other points below

K893 is a Psi-pivotal cubic, the locus of M such that X(110), M, Psi(M) are collinear. See Table 60.

It is also the locus of contacts of tangents drawn through X(3) to the circles passing through X(2) and X(110).

K893 is a focal cubic with focus X(3), with real asymptote the Fermat axis and its intersection with the cubic is X(6).

K893 is an isoptic cubic : for any point M on the curve, the directed line angles (MX3, MX2482) and (MX6, MX542) are equal (mod. π).

The tangent at X(2) is the Euler line. The tangent at X(3) is the Brocard axis.

The polar conic (C) of X(3) is the circle passing through X(2), X(3), X(110), X(842).

The polar conic of the infinite point X(542) is the rectangular hyperbola (H) passing through the midpoint X(5642) of X(2)X(110) whose asymptotes are the Fermat axis and its perpendicular at X(3).

The polar conic of X(110) is the rectangular hyperbola passing through X(110) and the four foci the Steiner inellipse.

(H) meets the bisectors of the lines X(3) X(6), X(3) X(542) at the centers of anallagmaty. Only two are real namely S1, S2. These bisectors are the parallels at X(3) to the asymptotes of the Jerabek hyperbola or, equivalently, they are the axes of the inconic with center X(3).

The lines F1S1, F2S2 meet at T1 and the lines F1S2, F2S1 meet at T2. These two points lie on K893 and they are Psi-inverses, hence collinear with X(110). This same property is also true for the corresponding imaginary points of K893 which lie on the other axes of the inconics with centers X(2), X(3).

In other words, the pairs of axes of the inconics with centers X(2), X(3) can be seen as two decomposed rectangular hyperbolas (H2), (H3) which generate a pencil containing a third decomposed rectangular hyperbola (H110). This has center X(110) and meets K893 again at four points namely T1, T2 and two imaginary other points all lying on the polar conic of X(2). See figure below.

This property remains true for any Psi-pivotal cubic with singular focus on the Thomson cubic K002 hence with pivot on Q119, the Psi-transform of K002.

K893a