One way in which eastern orthodoxy, roman catholicism, and protestantism are similar is that each

1) Catholic Church (there is no Roman Catholic Church, that is a often used but erroneous name) and SOME Eastern Orthodox, which recognize the primacy of the Bishop of Rome but lead their own particular "state" churches themselves. They are in constant communication with Rome. This is not your answer. 

2) Some Protestants reject parts of the Old Testament as part of the Bible. No Catholic or Eastern Orthodox reject any of the Old Testament. This is not your answer. 

3) This is your answer. The Catholic Chuch and Eastern Orthodox share the same trunk of the tree, and Protestantism are the branches, many of which are falling off the tree. 

4) Byzantium was created as the eastern seat of the emporer Lycinius, who was co-emporer of the Roman Empire with Constantine. The Catholic Church was the official religion. When Islamic forces took over most of that eastern half of the Roman Empire, the Catholic Church could not communicate or effect diplomacy with Moslem rulers because Islam requires Sharia'a Law for governance. No one used the word "Byzantine" until the 18th Century. It was a word invention of Enlightenment thinkers. The words used for this region was "the Roman Empire" because the east half is all that survived the barbarian invasions by Goths, Germanic tribes, etc etc etc. Eastern Orthodox didn't form until much much later, and it formed because in places bordered by or rule by Islamic forces, Christianity was only allowed by these Moslems within the boundries of a given state. So you have the various churches of Eastern Orthodox: Armenian Orthodox, Syrian Orthodox, Egyptian Orthodox (Coptic), Greek Orthodox. All of these say "orthodox" to mark their unity with the Catholic Church. They have a long tradition of governing themselves because the popes could not govern churches in Muslim controlled hands or in areas inaccessible to them (Greece). But what we call the Byzantine Empire was Catholic and it was nearly taken over by heresies like the Mynophosites. Orthodox were non-Mynophosites, people who adhered to the Catholic Church.