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X(1), X(3), X(20), X(170), X(194)

excenters

six contacts of the Steiner deltoid of IaIbIc with the bisectors

three cusps of this Steiner deltoid

points at infinity of the McCay cubic

more points and details below

Kp60 = Kp(X6) = K077 is the locus of pivots of isogonal pK+. The locus of the common point of the three asymptotes is Kc60 = Kc(X6) = K078.

This cubic is cited in several articles by R. Deaux, Mathesis 1959. See also the related cubic K609.

Kp60 is a member of the class CL014 of cubics.

Kp60 is a K60+ with three real asymptotes parallel to the sidelines of the Morley triangle and concurring at X(376) midpoint of X(2)X20).

The tangent at O is the Euler line. The tangents at the in/excenters concur at K = X(6).

K077, K078, K100, K258 are members of a same pencil of cubics generated by the McCay cubic K003 and the decomposed cubic which is the union of the Stammler hyperbola and the line at infinity. See Table 63.

K077 contains:

• S1, S2, S3 : cusps of the deltoid, on the circle C(O,3R). S1S2S3 is obviously equilateral.

• T1, T2, T3 : points on the Darboux cubic, on the circle C(O,3R)

• L1, L2, L3 : pedals of L = X(20) on the sidelines of S1S2S3

• Q1, Q2, Q3 : cevians of O in T1T2T3.

Note that the six points Q1, Q2, Q3, L1, L2, L3 lie on a same conic. See below.

• six points on the bisectors of ABC, on the deltoid, on the lines homothetic of the perpendicular bisectors of ABC under the homotheties with ratio 2, centers A, B, C. These points are two by two collinear with L.

***

K077 is the McCay cubic of the triangle T1T2T3.

The homothety h(O,-3) transforms K077 into K078.

***

Let (P) be the poloconic of the line at infinity in the isogonal pK with pivot P.

K077 is the locus of P such that (P) is degenerate.

C(O, 3R) is the locus of P such that (P) is a rectangular hyperbola.

The deltoid is the locus of P such that (P) is a parabola. When P lies inside, outside the deltoid, (P) is an ellipse, a hyperbola respectively.

It follows that, when P is one of the cusps of the deltoid, the pK is a trident with a triple point at infinity.

K077 is a pivotal cubic with pivot L

The tangential of O is L = X(20) and the tangential of L is X(194) = K/G (cevian quotient).

It follows that the polar conics C(L) and C(K/G) of L and X(194) have already three known common points with K077 which are precisely :

• L (twice) and X(3) for C(L),
• K/G (twice) and L for C(K/G).

Each polar conic must meet K077 at three other points : these are T1, T2, T3 for C(L) and Q1, Q2, Q3 for C(K/G).

From this, we see that the triangle Q1Q2Q3 is the diagonal triangle of the quadrilateral OT1T2T3.

Thus K077 is also the pivotal cubic with pivot L, isopivot X(194) with respect to Q1Q2Q3. This confirms that :

• the tangents at T1, T2, T3 pass through L,

• K077 meets the sidelines of Q1Q2Q3 at the cevian points R1, R2, R3 of L,

• the tangents at Q1, Q2, Q3 pass through X(194),

• the isoconjugate of the line at infinity is a circum-conic in triangle Q1Q2Q3 which must contain L1, L2, L3, the isoconjugate points of the infinite points of K077. Hence the lines LL1, LL2, LL3 are parallel to the asymptotes of K077.