Given an acute triangle ABC . It's altitudes AA1,BB1 and CC1 intersect at a point H , the orthocenter of △ABC. Let the lines B1C1 and AA1 intersect at a point K, point M be the midpoint of the segment AH. Prove that the circumscribed circle of △MKB1 touches the circumscribed circle of △ABC if and only if BA1=3A1C.(Bondarenko Mykhailo) geometrycircumcircletangent circlesatltitudesorthocenter