Let A1, B1, C1 be the apices of the external Fermat triangles; A', B', C' the altitude feet; A*, B*, C* the reflections of A, B, C on A', B', C'.
1) The circles {A*B1C1}, {B*C1A1}, {C*A1B1} have a common point
2) The circles {A1B*C*}, {B1C*A*}, {C1A*B*} have a common point