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K912

too complicated to be written here. Click on the link to download a text file.

X(3), X(15), X(16), X(23), X(110), X(5663)

inverses in (O) of X(5463), X(5464) : these are X(13858), X(13859) now in ETC (2017-07-13)

vertices of the Thomson triangle

K912 is the inverse in the circumcircle (O) of K463. It is a focal cubic with singular focus X(23), the inverse of the centroid X(2).

The orthic line is X(3)X(110) with infinite point X(5663), hence K912 is the locus of contacts of tangents drawn through X(23) to the circles passsing through X(3) and X(110). In particular, the polar conic (C) of X(23) is the circle passing through X(3), X(23), X(98), X(110), etc.

The real asymptote is the line X(323)X(3663), the image of X(3)X(110) under the homothety h(X23, 2).

K912 is invariant under the usual involution f on a focal cubic which maps a point M to the center of its polar conic with respect to its stelloidal prehessian.

f swaps (O) and the circle with diameter X(3)X(2930) which contains the homologues R1, R2, R3 of the vertices Q1, Q2, Q3 of the Thomson triangle.

K912 is related with stelloidal pentagonal circum-quartics. See Q125 and specially Q132.