1- Stellar aberration.
2- Stellar parallax.
3- The Doppler Effect.
4- Retrograde motion of planets.
5- Phases of Venus.
The debate between geocentism and heliocentrism has a long history going back at least to 300BC. The reason for the long debate is because mathematics of the times were unable to completely explain either model.
Most European churches in 200BC were in favor of believing that God had placed Earth in the center of the Universe. In 150AD The Roman Catholic Church chose to adopt the geocentric model because Claudius Ptolemaeus offered the most mathematical support for the geocentric model and it did not conflict with Biblical stories. The Church then chose to condemn anyone who spoke or wrote about heliocentrism unless they spoke of it as a mathematical theory only. The unfortunate problem was that Ptolomy had only the use of Euclidean geometry as the source of his computations which led to many errors and false misconceptions which then lasted for the next 1500 years. And religious denial lasting through 2008AD.
With the advancement of observations and mathematical models, the arguement reached fanatical stages in the 1500s when the Roman Catholic Church felt that heliocentrism was a threat to the Church's power and many supporters of the heliocentric models were imprisoned, tortured, and killed by hanging or by being burned alive in public.
The advent of the telescope in 1609 magnified the problems of geocentrism. The Church decided that one could speak about heliocentrism only as a mathematical exercise and not as a true fact. Books about heliocentrism were not allowed unless the Church had a chance to make 'corrections' and edits before publication.
Johannes Kepler, after analysing Tycho Brahe's observations, constructed his three laws in 1609 and 1619, based on a heliocentric view where the planets moves in elliptical paths. Using these laws, he was the first astronomer to successfully predict a transit of Venus for the year 1631.
In 1687, Isaac Newton devised his law of universal gravitation, which introduced gravitation as the force that both kept the Earth and planets moving through the heavens and also kept the air from flying away, allowing scientists to quickly construct a plausible heliocentric model for the solar system.
In 1725 James Bradley discovered stellar aberration. This is apparent yearly change in positions of all stars in the sky due to Earth's own motion. Aberration arises due to adding up of the speed of light coming from the star and Earth's own speed. This requires complex mathematics.
The Catholic Church lifted the ban on heliocentism books in 1757 because the Church could no longer keep up with imprisoning, torturing, or killing the growing number of persons believing in heliocentrism. It wasn't until 1822 that heliocentrism books could be written in Rome.
Stellar parallax is too small to be detected with the human eye. This fact was the main glue that held the geocentric models together. In 1838 astronomer Friedrich Bessel successfully measured the parallax of star 61 Cygni disproving that parallax did not exist. This also proved that stars were a very great distance away and not held within a circular epicycle which sped around Earth once each day. Parallax made the need for multiple epicycles obsolete.
In the mid 1800s it was proven that our Sun was a star like the others in the sky.
Since 1988, over 300 exoplanets have been discovered orbiting other stars. Never has a star been detected orbiting a planet.
Geocentric models had multiple difficulties explaining orbits. As a last attempt, they had planets going around the Sun but the Sun still went around the Earth.
http://en.wikipedia.org/wiki/Image:Tychonian_system.svg
Wavelength of the light that we receive from objects moving relative to us becomes a little shorter (bluer) when we approach the source and becomes longer (redder) when we move away from the source. The Doppler Effect. When Earth moves toward a star, the star will appear slightly bluer (only high-tech instruments can measure this) while it will appear redder when Earth is on the other side of the orbit and moves in the opposite direction. This effect proves that Earth has a velocity relative to the stars, similar to aberration.
The Coriolis force proves that the Earth is spinning and moving through space.
Space travel of probes has confirmed beyond any possible doubt the order of the planets and the fact that they all revolve around the central Sun. The mission of space probes depends upon highly advanced mathematical computations to fly past or land upon planets and moons in our Solar System. The order of the Solar System with the Sun in the middle cannot be denied with todays technology.