Points being the orthocenter of their cevian triangle

1) Let A', B', C' be the midpoints of the sides of ABC. Let B", C" be the projections of A on A'C' , A'B'.
2) Draw the hyperbola Ha through A, A', B", C" with asymptotes parallel to BC and OA, O being the circumcenter.
3) Draw analogously the hyperbolas Hb, Hc. The four common points to the three hyperbolas are the requested points.
The figure on the right shows the property.