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X(4), X(6), X(112), X(523), X(35906), X(35907), X(35908), X(35909)

vertices of the 4th Brocard triangle A4B4C4

other points below

Geometric properties :

K1143 is the isogonal transform of the Brocard (sixth) cubic K022.

It is a circular cubic with singular focus X(9970), the reflection of X(67) in X(5).

K1143 is invariant under the involution Psi2 mentioned in page K1142 and it is (apart K018) the only invariant circum-cubic.

K1143 also contains :

• E0 = X(35909) = isogonal conjugate of X(7473), the tangential of X(523) hence on the real asymptote, also on the Jerabek hyperbola,

• E1 = X(35906) = Psi2(E0), on the lines {4, 112}, {6, 523},

• E2 = X(35907) = X25 ÷ E0, on the lines {4, 6}, {112, 523},

• E3 = X(35908) = Psi2(E2) = X25 ÷ E2, on the lines {4, 523}, {6, 112}.

The isogonal conjugates X(35910), X(35911), X(35912) of E1, E2, E3 obviously lie on K022.