Equicenter of two inscribed triangles

1) Let A'B'C', A"B"C" be the two inscribed triangles, D'E'F', D"E"F" their antimedial triangles.
2) A'B', A"B" intersect in C*; construct analogously B*, A*.
3) D'E', D"E" intersect in F*; construct analogously E*, D*.
4) The lines C*F*, B*E*,  A*D* intersect at the equicenter E.