### 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_{t}C = CB_{t} : B_{t}A = AC_{t} : C_{t}B = 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**_{t}E_{t}F_{t} have the same centroid, for all t.

http://amontes.webs.ull.es/otrashtm/HGT2017.htm#HG030917

Angel Montesdeoca. Sep, 2017