X(3), X(6), X(25), X(55), X(56), X(64), X(154), X(198), X(1033), X(1035), X(1436)
vertices of the tangential triangle
vertices of the Thomson triangle
isogonal conjugates of the CPCC or H-cevian points, see Table 11
The inversive image of K172 in the circumcircle is Q071.
Let ABC be a triangle, P a point, and PaPbPc the pedal triangle of P. La, Lb, Lc are the parallels to BC, CA, AB, resp., through P. Denote by : Lab = La /\ AB, Lac = La /\ AC, Ab = the orthogonal projection of Lab on Lc, Ac = the orthogonal projection of Lac on Lb, A' = BAb /\ CAc, and similarly define B', C'. The locus of P such that ABC, A'B'C' are perspective is the Thomson cubic. The locus of the perspector is K172. Note that the cubic is also a pK with pivot K, invariant under the isoconjugation with respect to the tangential triangle which swaps K and O. (Antreas Hatzipolakis, Jean-Pierre Ehrmann, Hyacinthos #8480-83-87)
Its asymptotic directions are those of pK(X6, X2979) where X2979 = a^2(b^4 + c^4 + b^2c^2 - a^2b^2 - a^2c^2) : : , intersection of the lines X(3)X(54) and X(2)X(51).
We met K172 in the solution of the Darboux problem : one considers the conics circumscribed to a triangle such that the normals at the three vertices of the triangle are concurrent at a point P. One asks the locus of the foot of the fourth normal passing through P (literal translation). In , it is shown that the locus is a 7th degree curve which is the isogonal transform of the inverse (in the circumcircle) of a non-identified cubic. It appears that this cubic is K172 and consequently the locus is the antigonal image of the Lucas cubic (see Hyacinthos #7825 & sq.). Antigonal is defined here. We call this locus the Darboux septic or Q001.
 André Haarbleicher: De l'emploi des droites isotropes comme axes de coordonnées. Nouvelle géométrie du triangle. Paris, Gauthier-Villars, 1931, pp. 61-70. See also Hyacinthos #8509-11-12-15.