Angle-chasing usually works:
$\displaystyle \angle OBA = 90-\alpha$
$\displaystyle \angle DBC = 90-\alpha$ (due to symmetry)
$\displaystyle \angle ABC = 2 \alpha$ (since OBA + DBC + ABC = 180 deg)
$\displaystyle \angle ACB = 90 - 2\alpha$
$\displaystyle \angle EC(D) = 90 - 2\alpha$ (also due to symmetry)
$\displaystyle \angle BC(D) = \beta = 4 \alpha$ (since ACB + EC(D) + BC(D) = 180 deg)