> Let D1, E1, F1 be the projections on the xy-plane of > the respective touching points of the half-circles > with the tangent plane. The lines DD1, EE1, FF1 are > concurrent at the symmedian point X(6) of triangle > DEF. Furthermore, the intersection of the tangent > plane with the xy-plane coincides with the tripolar > line of the symmedian point.
I have verified independently that DD1, EE1, FF1 are concurrent in the general case. You have gone quite a bit beyond in identifying the intersection point and the trace of the tangent plane. Good job!