22-X(15) or X(16) and three antipode lie on a circle

Let  be a triangle. Construct three equilateral triangle on the side:  ( either all outward, or all inward).

Construct circle through , and through midpoints of ,, center of circle is 
Construct circle through , and through midpoints of ,, center of circle is 
Construct circle through , and through midpoints of ,, center of circle is .

Prove that:
- is antipode of  with 
 is antipode of  with 
 is antipode of  with 
Four points:  or  and A',B',C' lie on a circle