DOHA, Qatar — The Arab uprisings have appeared to offer a new bridge in the anguished relationship between the United States and the Muslim world, perhaps most of all in Libya, where American firepower helped oust Col. Muammar el-Qaddafi and earned the United States deep popular gratitude.

A history of mutual suspicion seemed to be giving way to a shared language of civic rights and freedoms; American visitors are often still greeted with smiles and hugs.

But on Wednesday, in the wake of angry anti-American protests and the killing of the American ambassador and three other diplomatic workers in Benghazi, Libya, there was a return to sadly familiar ground. The cause, as in earlier riots that followed the burning of Korans in Afghanistan and the publication of anti-Islamic cartoons in Denmark, was a perceived blasphemy — in this case a video clip that denigrated the Prophet Muhammad.

Once again, Muslims were furious, and many in the West found themselves asking why Islam seems to routinely answer such desecrations with violence.