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K1141

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X(2), X(6), X(111), X(523), X(5653), X(5912), X(5968), X(14899), X(35607), X(35608), X(35609)

vertices of the Thomson triangle

Z3 = Psi(X5912) = X(35606) on {2,111}, {6,523}

Geometric properties :

When R ≠ X6, the orthic line (L) of a nK0(X6, R) always passes through X6. Hence the polar conic of X6 is a rectangular hyperbola with center Ω = Psi(R).

Recall that nK0(X6, X6) is the stelloid K024 and the polar conic of every point R is a rectangular hyperbola with center Psi(R). See Orthopivotal cubics.

Ω lies on (L) if and only if R lies on the circular cubic K1141 with singular focus X(11579).

Since Psi is an involution, K1141 is a Psi-cubic as in Table 60.

K1141 is an isogonal pK in triangle X(2)X(6)X(111) with pivot X(523) and isopivot X(5653). It follows that K1141 must pass through the in/excenters of X(2)X(6)X(111) which lie on the axes of the inscribed Steiner ellipse with real foci F1, F2 and on the axes of the orthic inconic. The incenter is X(14899) and the excenters are X(35607), X(35608), X(35609).

Note that X(5653) lies on the Jerabek-Thomson hyperbola (JT), on the circumcircle (C) of X(2)X(6)X(111) which also contains the singular focus X(11579).

The polar conic (H) of X(5653) is the hyperbola passing through X(2), X(6), X(111), X(523), X(5653).

X(5653), X(5912), X(5968), Z3 share the same tangential T4, a complicated point on K1141. Hence K1141 is also a pK in triangle X(5912)X(5968)Z3 with pivot X(5653) and isopivot T4.