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The answer to this would have been different at different points in the thousand-year period that you are asking about. For the last two hundred years or so of this time period, Western Europe clearly won out over the others and was much more powerful than they were in 1800. However, there would have been many times earlier in the period that one or both of the other cultures would have been better off.
For example, in the early 1400s, China was extremely powerful. It was able to send out Admiral Zheng He in huge ships to make voyages that went as far as the east coast of Africa. As another example, the Muslims in the Middle East were strong enough in the early 1500s that Suleiman the Magnificent was able to take much territory in the Balkans and besieged Vienna.
However, as time went by, the Western Europeans progressed much more than the other cultures did. By 1800, the Europeans had a huge technological advantage over the other groups. Their guns and other such technology allowed them to dominate the others. This means that, by the end of the period you are asking about, the Europeans were clearly ascendant.
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