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X(2), X(3), X(110), X(525), X(7473)

vertices A2, B2, C2 of the second Brocard triangle

their X(3)-isoconjugates A'2, B'2, C'2 : these points are the intersections of the parallels at X(2) to the sidelines of ABC with the corresponding cevian lines of X(69). They lie on the circle with diameter X(2)X(376). Their coordinates are (a^2 : 2 SB : 2 SC), etc.

feet A', B', C' of the trilinear polar of X(524) = infinite point of X(2)X(6).

other points below

This cubic is met in "Orthocorrespondence and Orthopivotal Cubics" (end of §5). K022 is a Psi-cubic as in Table 60.

K022 is a circular nK with pole X(3) and root X(524). Its isogonal transform is K1143.

K018 and K022 are invariant under the involution Psi described in the paper above. See also Inverses of Isocubics.

K022 is the isogonal pK with pivot X(525) in the triangle T with vertices G, O, X(110). It follows that :

• K022 contains the in/excenters of this triangle T. Note that the bisectors at G are the axes of the Steiner inellipse and that those at O are the axes of the inconic with center O hence they are parallel to the asymptotes of the Jerabek hyperbola. This gives another construction of these four in/excenters. Naturally, the tangents at these points are parallel to the real asymptote.

• the intersection X of K022 with its real asymptote is the isogonal conjugate of X(525) with respect to T.

• the singular focus F of K022 is the antipode of X on the circumcircle of T. Also, F = X(14649) = Psi(X6800).

• K022 must contain the traces of X(525) on the sidelines of T. These points form the diagonal triangle of the quadrilateral X(2) X(3) X(110) X(525). In particular, X(7473) is the third point on the Euler line.

The involution Psi swaps :

• A, B, C and the vertices A2, B2, C2 of the second Brocard triangle,

• A', B', C' and A'2, B'2, C'2,

• X(3) and X(110), X(7473) and the third point on X(2)X(110), X and the third point on X(3)X(110).

See K1142 where an analogous involution Psi2 is introduced.