If the feet of lines from a point P, which lines are parallel with the sides of a triangle ABC, are
collinear according to two different lines then P lies on an appropriate and unique circum ellipse and vice versa.

Instead of proof:
The statement apparently true if ABC triangle is equilateral and the appropriate ellipse is the circumcircle.
Everyone can create a linear (affine) transformation which makes equilateral triangle from an arbitrary triangle and an inverse (affine) transformation from an equilateral triangle to an arbitrary triangle. So the circumcircle above become to the appropriate Steiner circumellipse by the invers transformation and parallel lines remain parallel.
(See a geometric construction of Steiner circumellipse )