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X(67), X(99), X(316), X(523)

other points below

K091 is the only isotomic focal nK. See Special isocubics ยง4.3.6. It is a member of the class CL028 and it belongs to a pencil described in the page K088.

It is the locus of M such that the circle with diameter MM' (M' isotomic conjugate of M) passes through the Droussent pivot X(316).

The singular focus is X(67), isotomic conjugate of X(316). The root of the cubic is X(11056), the orthocorrespondent of X(316).

The real asymptote is perpendicular to the Euler line and meets the curve at X, intersection of the perpendicular at X(316) to X(316)X(67) and the perpendicular at X(99) to X(99)X(67). These four points are therefore concyclic. The isotomic conjugate Y of X is the intersection of the line X(316)X(67) and the perpendicular at X(99) to the Euler line. The polar conic of X passes through X(67), X(99), X(316), X(523).

An easy construction of K091 is the following : draw a circle passing through X(99), X(316) and the line through its center and X(67). They intersect at two points on the cubic. See another construction and a generalization in CL058.

This yields that K091 is also the locus of contacts of the tangents drawn through X(67) to the circles centered on the line X(99)X(316) and orthogonal to the circle with diameter X(99)X(316).



K091 is an isogonal focal pK with pivot X(523) with respect to the triangle T = X(67)X(99)X(316).

Note that the line X(67)X(523) is an altitude of T hence X is the isogonal conjugate of X(523) with respect to T.

Naturally, K091 contains the four in/excenters of T and the traces of X(523) on T.


K091 has always three real prehessians K1, K2, K3 and one of them K1 is a stelloid i.e. a K60++ with radial center X(67), the singular focus of K091.

Each prehessian Ki is associated with a natural quadratic involution Fi on K091: for any point M on K091, let Fi(M) be the center of the polar conic of M with respect to Ki. Fi(M) lies on K091.

F1 maps the vertices of ABC to the traces U, V, W of the trilinear polar of the root R.

F2 and F3 maps ABC to two other triangles A2B2C2 and A3B3C3, each being the isotomic transform of the other.


  • ABC and A2B2C2 are perspective at X(1799),
  • ABC and A3B3C3 are perspective at X(427),
  • A2B2C2 and A3B3C3 are perspective at P, the harmonic conjugate of R with respect to X(1799) and X(427).

It obviously follows that the axis of perspective of A2B2C2 and A3B3C3 is the trilinear polar UVW of R.

Note also that A, B2, C3 and A, B3, C2 are collinear, the other points similarly.

These involutions have one real singular point namely X(99), X(316), X(67) respectively and two other imaginary points.

Those of F3 are the circular points at infinity and those of F1, F2 are the intersections of the trilinear polar of X(76) -- the orthic line of K091 -- and the isotropic lines passing trough X(99), X(316) respectively.

From this we see that F3 transforms any line into a circle passing through X(67). For instance, F3 transforms

  • the line X(99)X(316) into the circle X(67)X(99)X(316),
  • the line A1B3C3 into the circle BCA2X(67),
  • the orthic line of K091 into the polar conic (a circle) of X(67).

At last, let us recall that the poloconic of any line L in one of the three prehessians is a conic tritangent to K091 at the homologues of the intersections of the line L with K091.

Taking L = UVW, we obtain three conics tritangent to K091 at the vertices of the triangles A2B2C2, A3B3C3 and ABC.