How did chritian crusades contribute to the decline of the byzantine empire?

In quite a number of ways. Initially, the Byzantine Emperor requested a "small force" of soldiers from Western Europe. Instead, Pope Urban II seized on the request as an opportunity to do a number of things. One, the Church wanted to ensure the safety of European pilgrims to the Holy Land. Two, the Church wanted very much to end the constant, bloody warfare between European knights. Three, the Church (by "Church" I'm referring to the Roman Catholic Church) wanted to exert supremacy over the Greek Orthodox Church of Byzantium, which had split from the Catholic Church centuries earlier. Four, the Church (and the fledgling nation states of mainland Western Europe) wanted to ensure that Islam, which had very nearly conquered Europe in the 8th Century (and still controlled Spain), could not threaten them again.

With those goals (and others), Pope Urban II launched the first of what would become several Crusades. The Crusader armies immediately set out for the Holy Land, mostly by marching through central Europe. Local communities were forced to supply the armies, and were often destroyed if they refused. Virulent anti-Semitic pogroms took place. When the Crusaders reached Constantinople (Istanbul), they had to be fed and housed. This put incredible strain on the Byzantine economy, which often collapsed under the burden, making famine a specter that haunted the Empire for centuries. Several times, hungry Crusader armies razed Constantinople. Additionally, the Byzantine Emperors and Empresses were often at the mercy of massive Crusader Armies and their mercurial commanders, who could (and did) depose the Byzantine government and install a new one if they chose to do so.

The Byzantines were expected to participate in the wars - and found their armies decimated by constant warfare. Without disciplined, standing armies, the Byzantine Empire was vulnerable to invasion. Furthermore, the constant influx of foreigners and foreign armies brought something else: disease. Smallpox, the Plague, even Influenza (the Flu) wrought terrible destruction on the Byzantine Empire before spreading to other nations.

Meanwhile, the Turks and Arabs of the region came to identify the Byzantine Empire as the cause of all their troubles. Certainly the Western Europeans - called Ferengi (which is how the locals pronounced the word "Frank", the original name of the French) - caused most of the strife, but when they left, the only ones remaining were the Byzantines... who found themselves bearing the brunt of Turk and Arab retaliation. Eventually the Ottoman Turks destroyed the Byzantine Empire and changed the name of its capitol from Constantinople to Istanbul.

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