Geometry Problem 482: Triangle, Circumcircle, Incenter, Excenter, Midpoint, Cyclic Points

The figure shows a triangle ABC with circumcircle C₁, incenter D, and excenter E corresponding to BC. If F is the midpoint of arc BC, prove that points D, B, E, and C lie on a circle with center at F.
Circumcircle of a triangle is the circle which passes through the vertices. Incenter is the center of the incircle. Excenter is the center of an excircle.