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X(2), X(4), X(6), X(25), X(111), X(112), X(125), X(1560), X(5622), X(14899), X(35607), X(35608), X(35609), X(35901), X(35902), X(35903), X(35904), X(36201), X(36202), X(36203), X(36204)

other points below

Geometric properties :

K1142 is a circular cubic invariant under the involution Psi described in the page K018 and in the paper "Orthocorrespondence and Orthopivotal Cubics", §5. K1142 is a Psi-cubic as in Table 60.

The real point at infinity Z = X(36201) lies on the lines {X4, X1177}, {X6, X1562}, {X20, X1632}, {X25, X125}, {X30, X511}, {X64, X67}, {X66, X74}, {X110, X1370}, {X113, X206}, {X159, X2935}, etc, with SEARCH = 1.08671315321222.

K1142 is the isogonal pK with pivot Z with respect to the triangle T = X(2)X(6)X(111) hence it must contain the in/excenters X(14899), X(35607), X(35608), X(35609) of T. Note that these four points lie on the axes of the Steiner inellipse and the orthic inconic.

The isogonal conjugate X = X(36202) of Z in T is the intersection with the real asymptote and the antipode F of X in the circumcircle of T is the singular focus. Note that X is the common tangential of X(2), X(6), X(111) and Z.

K1142 meets the sidelines of T again at Q1 = X(36203), Q2 = X(35903) , Q3 = X(35904) on the lines {X2, X6}, {X2, X111}, {X6, X111} and obviously on the parallels at X(111), X(6), X(2) to the asymptote respectively. Note that Q2 = Psi(Q1) and Q3 = Psi(X).

K1142 also passes through :

• P1 = X(35901) = Psi(X112) on the Brocard circle and on the lines {X3, X647), {X6, X25}, {X111, X5622}, {Z, X5622},

• P2 = X(35902) = Psi(X1560) on the lines {X4, X6}, {X111, X125}, {Z, X112},

• Q4, Q5 on the circumcircle (O) and on the line {X125, X468},

• Q6 = X(36204) on the lines {X4, X112}, {X125, P1},

• P4 = Psi(Q4), P5 = Psi(Q5) on the Brocard circle, on the line {X125, X15000}, on the circle GHX(6776) where X(6776) is the reflection of H in K,

• Q7 on the line {X4, X125} and on the circle X(4), X(23), X(148), X(895),

• Q8 on the line {X112, X125}.

***

K1142 is also invariant under another involution Psi2 very similar to Psi where the Steiner inellipse is remplaced by the orthic inconic. Psi2(M) is the commutative product of a reflection in one axis of the orthic inconic and the inversion in the circle with diameter the foci of this same conic. Psi2(M) may also be seen as the center of the polar conic of M in the stelloid K598, with radial center X(6) and asymptotes parallel to those of K024.

With M = u:v:w, this point Psi2(M) is : a^2(4 SB SC v w - 2 c^2 SB v^2 - 2 b^2 SC w^2 + b^2 c^2 u (-u+v+w)): : .

In particular, Psi2(X6) = Z, Psi2(Q7) = Q8.

The only circum-cubics invariant under the involution Psi2 are K018 and K1143, the isogonal transform of K022.

See here for a list of pairs {P, Psi2(P)} and a generalization with transformations Psi_P.