A problem related to Artzt parabolas

Hyacinthos #26563, (Antreas Hatzipolakis)

Let ABC be a triangle and t a real number.
A_t, B_t, C_t such that BA_t : A_tC = CB_t : B_tA = AC_t : C_tB = t.
D_t is the intersection of the A-Artz parabola with the parallel by A_t to its axis.
E_t is the intersection of the B-Artz parabola with the parallel by B_t to its axis.
F_t is the intersection of the C-Artz parabola with the parallel by C_t to its axis.

Then the triangles ABC and D_tE_tF_t have the same centroid, for all t.

http://amontes.webs.ull.es/otrashtm/HGT2017.htm#HG030917
Angel Montesdeoca. Sep, 2017