Four points on Euler Line

I introduce  four new points on Euler line:

I found these points trying to solve a problem proposed by Kadir Altintas.

Given a triangle ABC and a point P, call DEF its circuncevian triangle. The circumcenters of six triangles PBD, PDC, PCE, PEA, PAF and PFB lie on the same conic. When this conic is a circle?

Updated March, 1. Additions/corrections are in red.

Four points on Euler line