Let ABC be an acute triangle with orthocenter H, circumcenter O, and circumcircle Ω. Points E and F are the feet of the altitudes from B to AC, and from C to AB, respectively. Let line AH intersect Ω again at D. The circumcircle of DEF intersects Ω again at X, and AX intersects BC at I. The circumcircle of IEF intersects BC again at G. If M is the midpoint of BC, prove that lines MX and OG intersect on Ω. geometryorthocentercircumcircleCircumcenter