Here's the proof word by word:

"Consider a point P' in 3D such that PP' is perpendicular to the plane of triangle ABC. Let points X', Y', and Z' be symmetries of P' in the midpoints of CB, BA, and AC, respectively. Then points A, B, C, P', X', Y', Z' serve as vertices of a parallelepiped. The required concurrence of AX', BY' and CZ' is obvious. Now observe that the original configuration is the orthogonal projection of that parallelepiped onto the plane. The conclusion follows since projection preserves concurrence."

One thing I can add is that, by construction, XX', YY', ZZ' are perpendicular to the plane of triangle ABC. Other than that I just do not know what else to say. You may see something that I do not. The above paragraph consists of 6 sentences. Which one causes the trouble?

Bets regards,
Alexander Bogomolny