It didn't, at first. Copernicus was no better at predicting the exact position of the planets relative to Earth than was Ptolemy. This was because Copernicus was still relying on circular, uniform motion. Only when Kepler threw out circles for Ellipses with varying speeds did the predictions work.
Still, this was too subtle and mathematical for many Geocentric believers. The nail in the coffin was Gaileo's telescope, which showed "flaws" in the moon's surface, other moons orbiting Jupiter (not Earth), and Venus having phases that could only be explained by a sun-centered orbit.
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Geocentric theory
Geocentric theory
The heliocentric theory is the idea that planets rotate around the sun, where the geocentric theory said that everything orbited around Earth.
The Geocentric or Ptolemaic Model put the earth at the center of the Universe. The Heliocentric Model postulated by Copernicus and, before him, Aristarchus, places the Sun at the center of the Solar System. Galileo's observations proved the validity of the Heliocentric Model.
Heliocentric = The Sun is at the center of our solar system. Geocentric = The Earth is at the center of our solar system. Insisting the Sun is at the center, which it is, once very much angered the church.