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X(2), X(99), X(523), X(34205), X(34206)

This is the only circular isotomic cK with node at G. See Special isocubics §8.5.2.

Its root is X(11054), on the line X(2)X(39).The singular focus is X(14653).

K088 is the Psi-image of the conic passing through G and the vertices of the second Brocard triangle. The tangent at G is parallel to the lines {74,98}, {99,110 }, {113,114}, {115,125}, etc.

Psi is the involution that is the product of a symmetry about one axis of the Steiner inellipse and the inversion in the circle with diameter F1F2 (the foci of the ellipse).

More details in "Orthocorrespondence and Orthopivotal Cubics", §5.

See also K394 and K018.

Locus property

Let G, K, S be the points X(2), X(6), X(99) in ETC and let M be a variable point on the line GK. The circumcircle of GSM meets the circum-conic of ABC passing through G and M at G (counted twice since the tangent is the same), M and then another point P. When M traverses GK, the locus of P is K088 (Angel Montesdeoca, 2020-04-06).

More generally, if S is any point on the circumcircle of ABC, the locus of P is another circular nK passing through G, S and X(523). See K1153 for further details.

When S = X(110), the cubic is the isogonal focal nK(X6, X22329, X2) with singular focus X(110). See also the related CL061 and K397.

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K088 is a member of the pencil of isotomic nKs generated by the two following decomposed cubics :

• union of the Steiner ellipse and the line at infinity,

• union of the circumcircle and the de Longchamps line, the trilinear polar of X(76) and the isotomic transform of the circumcircle.

Each cubic is circular with singular focus on the line X(3), X(67), etc. The root lies on the line X(2), X(39), etc.

Special cubics of the pencil :

• K088 which is a cK with node X(2) and three other cKs with nodes the vertices of the antimedial triangle.

• K091 which is a focal cubic.

• K197 which is the only nK0 of the pencil.

• an axial cubic (K) with

• singular focus X(8724),
• asymptote the line X(2)X(523),
• axis of symmetry the line X(30)X(99),
• root the point R =X(22254) on the lines {2,39}, {30,17941}, {316,5468}, {671,1641}, {868,7809}, {1989,18896}, {5108,7790}.

(K) meets its axis of symmetry at X(99) and two other points M1, M2 which are isotomic conjugates hence lying on the conic (C), the isotomic transform of its axis.

(C) is a circum-hyperbola passing through X(67), X(98), X(523), X(542), X(1494), X(1989).

Obviously, the tangents at X(99), M1, M2 are parallel to the asymptote hence perpendicular to the Euler line.