In the figure below, circles O1, O2,
and O3 are tangent to each other and to a line L at A, B, and E,
respectively. The circle O4 passes through A and B and is
tangent to circle O3 at F. Tangent to the circle O4 at the
midpoint of major arc AB meets O1A and O2B extended at D and C,
respectively. Prove that ABCD is a square.