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X(1), X(12), X(65), X(85), X(14584)

vertices of intouch triangle

infinite points of pK(X6, X2975)

Geometric properties :

K1058 is psK(X1254, X7, X1) in Pseudo-Pivotal Cubics and Poristic Triangles, see Table 5 in particular and also K360 = psK(X56, X7, X1).

K1058 is a nodal cubic with node X(65) and nodal tangents parallel to the asymptotes of the circum-conic with perspector X(661), passing through X(1), X(10), X(19), etc.

The common points Q1, Q2, Q3 of K1058 and C(O,R) are those of pK(X6, X20612).

For any point M = u : v : w ≠ X(37), the point M' = (b(u-v+w) - c(u+v-w))/((-a+b+c) (c(a+b)v - b(a+c)w)) : : is a point on K1058 which gives a lot of reasonably simple points on the cubic.