Concurrent Brocard Axes

Let $ABC$ be a triangle, $P$ a point and $A' = AP \cap BC$. If the Euler lines of three triangles from the set $\{A'AB, A'BP, A'PC, A'CA\}$ are concurrent, then the Euler lines of all four triangles are concurrent (see Forum Geometricorum 2001).

Antreas Hatzipolakis asks for the same for Brocard axes (see Hyacinthos message 20664). Let's investigate it using barycentric coordinates and Mathematica.

Concurrent Brocard Axes