Given that ABC is a triangle where AB<AC. On the half-lines BA and CA we take points F and E respectively such that BF=CE=BC. Let M,N and H be the mid-points of the segments BF,CE and BC respectively and K and O be the circumcenters of the triangles ABC and MNH respectively. We assume that OK cuts BE and HN at the points A1 and B1 respectively and that C1 is the point of intersection of HN and FE. If the parallel line from A1 to OC1 cuts the line FE at D and the perpendicular from A1 to the line DB1 cuts FE at the point M1, prove that E is the orthocenter of the triangle A1OM1. geometrycircumcircleorthocenter