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K1142

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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. 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.

K1142a