Grebe-orthic perspector generalized

On Feb 15, 2003, Darij Grinberg published a new result (Grebe-orthic perspector) associated to the Grebe Triangle here. A generalization of this is given in this post.
For more information you can visit Angel Montesdeoca's Site (Jan 30, 2014). 

Let ABC be a triangle. Let AB = a, BC = b, AC = c.
Let x, y, z be three distances from the sides of triangle ABC to the sides of triangle A'B'C', such that x/a = y/b = z/c.

Prove that the triangle A'B'C' is in perspective with the orthic triangle of ABC.

The proof at AoPS is due to Luís González.