Imagine the mathematical gymnastics required to describe the motion of the other planets in a geocentric solar system.
Exactly.
Or consider the math required to describe the motion of a golf ball hit by a three iron, relative to the center of the galaxy.
Anytime we're making calculations of movement, we choose the origin we're measuring things from. We choose the origin that is most convenient. When we're calculating the flight of a golf ball, we assume the earth is stationary. When we're calculating the movements of the planets, we assume the sun is stationary. In most cases, we choose a non-rotating frame of reference. But sometimes, we do choose a rotating frame of reference.
It's all a matter of what we're interested in, what we're not, and what makes the math easiest to work with.
There are equations that will describe the motion of Mars, assuming the sun is the origin, and that it rotates relative to the frame, and there are equations that will describe the motion of Mars assuming that the Earth is the origin and that the universe rotates relative to it.
Neither is more or less true than the other, it's simply a matter of which is more convenient.
Imagine the mathematical gymnastics required to describe the motion of the other planets in a geocentric solar system. (or would it be a terra system?)
The math shouldn't be that difficult. Assuming we know the other bodies actually orbit around the Sun, and we're simply making Earth the fixed reference point, you should just have a pretty simple ellipse for planets farther out than Earth. The orbital math would stay the same, you just have to add in the rotational motion of the Sun around the earth. The center of each planet's orbit is simply making it's own circle around Earth. So, most planets would just have a slightly bigger elliptical path (far enough that I don't think they'd create the loops like below, but I didn't do any math on it!). Planets closer to the sun than the Earth would be different, as their revolution is contained inside the Earth-Sun circle. Their path would look more like a continuous looping path like so:
