Points with equilateral cevian triangle

1) Let X, Y, Z  be the extra vertices of the external Fermat triangles.
2) Let A', B', C' the reflections of A, B, C with respect to the opposite sides.
3) The lines A'Y, A'Z intersect AB, AC at Ab, Ac respectively. Construct analogously Bc, Ba, Ca, Cb.
4) The conics {BCA'AbAc}, {CAB'BcBa}, {ABC'CaCb} intersect at most at three real points having equilateral cevian triangle.
The same construction with the internal Fermat triangles gives three more points (figure c65_2; only showing the cevian triangle of one of the three points).