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X(4), X(5), X(6), X(25), X(51), X(52), X(53)

vertices of the orthic triangle

midpoints of the orthic triangle

all the related points of the Thomson cubic with respect to the orthic triangle

K350 is the Thomson cubic of the orthic triangle. It is the pK with pole X(3199), pivot H, invariant in the isoconjugation which swaps H and X(51), the centroid of the orthic triangle.

It has the same asymptotic directions as pK(X6, X2979).

It is the isogonal transform of K646 = pK(X97, X95).

K350 is the locus of the pseudo-poles Ω of the stelloids psK60+ which have their asymptotes parallel to those of the McCay cubic K003. The corresponding pseudo-pivots P and pseudo-isopivots P* lie on K045 and K350 respectively.

More precisely, for any Ω on K350, the cubic psK(Ω, P, X4) where P is the anticomplement of the barycentric quotient X(51)÷Ω, is one of these stelloids.

The radial center X lies on the image of K044 under the homothety h(X5, 1/3).

For example, with Ω = X(4), X(5), X(6), X(25), X(51), X(53), X(14593) we find the cubics K028, K071, K003, K1140, K026, K049, K670 respectively.

See the file McCay Stelloids for further explanations.


Locus property (Angel Montesdeoca, 2022-04-25)

Let DEF be the cevian triangle of a point P, the circumcircle of triangle ADB intersect AC again at Ab and the circumcircle of triangle ADC intersect AB again at Ab. Let Oa be the circumcenter of triangle AAbAc. Define Ob, Oc similarly. The lines AOa, BOb, COc concur (at Q) if and only if P lies on K350. The locus of Q is K919.