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X(4), X(223), X(1439), X(3182), X(5930), X(5932), X(8807), X(8808), X(8809), X(8810), X(8811), X(8812)

vertices of the 3rd extouch triangle

other points below

Geometric properties :

For any point M on the Lucas cubic K007, the cevian triangle of M is perspective to the 2nd [resp. 3rd] extouch triangle and the locus of the perspector is K880 = pK(X37, X329) [resp. K963 = pK(X1427, X5932)]. See ETC, preamble of X(8782).

K963 meets the line at infinity at the same points as (K1) = pK(X6, P1) where P1 is the common point of the lines {1,29), {2,77}, {20,78}, etc. The remaining common points are A, B, C, X(223) and two others on the conic (C1) also passing through X(2), X(1103).

K963 meets the circumcircle at the same points as the cubic (K2) = pK(X6, P2) where P2 is a rather complicated point with SEARCH = -1.75657892397560. The three remaining common points lie on (L2) passing through X(6), X(282).