Constantinople wasn't part of the muslim world in the middle ages, it was the center of eastern orthodox christianity, it lost its fight with the western branch, and eventually was conquered by the muslims :) (well, depends on your definition of middle ages, but the opinion that middle ages ended with the fall of Constantinople is quite popular).
Europe yes, but power for the world as you suggest moved east, and remained in the Muslim world for most of the middle ages.
While people in London were dying in plagues, the folks in Constantinople were enjoying their new found success.