Which external enemy proved to be the greatest continuing threat to the byzantine empire in the 15th century

The greatest threat for the Byzantine Empire was the turkish invaders, who in 1453 would conquer the imperial capital, Constantinople, and give it it's modern name, Istanbul. The city had been conquered only once before in 1204 by catholic crusaders, and although it was reconquered in 1261 the Empire had been weakened and that allowed the turkish Ottoman Empire to grow in power, conquering Serbia and much of the Balkans in the 14th century, by the middle of the 15th century the Byzantine Empire was not much more than the city of Constantinople.

i don't get what you mean

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