I will do this as an exercise in argumentation.
I will prove that Earth is not flat, but round.
First, I prove that the distance(s) to the stars is very great compared to distances on Earth.
In your FAQ, you claim that the stars are situated 3100 miles above sea level. This is contardicted by the fact that the constellations always have the same apparent sizes, shapes and positions relative to each other, no matter where on Earth and when we observe them. As an example, take the so called Square of Pegasus. This consists of four stars that roughly form a square. Three of the stars are in the constellation Pegasus, the fourth in the constellation Andromeda. When the square reaches its highest postion in the sky, it's sides are roughly directed north/south and east/west. It reaches this highest position about midnight in September for locations at (what is usually called) the northern hemisphere. For a location in southern United States, this high position is a little bit south of zenith.
Now suppose that the square in visible in that position about midnight in September at location in Georgia or Florida. At the same time, it's about 9 p.m. in California. How will people in California then see the square? If we assume that all stars are located at same exact heights (ca 3100 miles) above sea level, as if they were located in a horizontal ceiling with at that height, Californians will not see the four stars making up a square, but a rectangle, located in east in the sky, with considerably longer north/south sides than east/west sides, but even the east/west sides are apparently shorter than the people in the east see them. This is because the direction to the square makes an angle of about 45 degrees with the vertical direction for the Californians, and the distance to the square is also about 1.4 times longer for the Californians than the people in the East. (Well ,a litlle bit less than 45 degress and the distance factor is a little bit less than 1.4, for the distance across the southern U.S. is a little less than 3100 miles, but that doesn't change the argument).
But this is in conflict with observation. The apparent size and shape of the square of Pegasus are the same wherever and whenerver we observe it on Earth.
But perhaps the distances to the stars vary a little about 3100 miles above sea level? This would make things even worse, because then the constellations would be even more distorted when we move from one location on Earth to another. The square of Pegasus then wouldn't even look like a rectangle for the Californians in the example above.
No, there is only one possible explanation of the fact that the sizes, shapes and relative positions of the constellations always look the same wherever and whenever they are observed on Earth: that the distances to them are very large compared to distances on Earth. It's like if you watch collection of objects about, say, 1 kilometer away from you, then the apparent relative positons of the objects doesn't change noticeably if you move a few meters in some direction.
So, the distances to the stars are very large compared to distances on Earth, so large that the sight lines to any particular star are almost parallell for any two observers on Earth.
But then, if Earth was flat, any star would be seen at the same height above the horizon for any two arbitrary observers on Earth watching the star simultaneously. But that is not the case. For example, the Pole Star is always seen at the same heitght above the horizon at every fixed loaction. In Scandinavia, where I live, it is seen quite close to the zenith. In southern Europe, and in the U.S., it is seen much closer to the horizon. This wouldn't be the case if Earth was flat. The only explanation is that the horizontal planes at different locations on Earth are not parallell, and therefore the Pole Star is seen at different heights at different locations (as does other stars also. I noticed, for example, that Orion is seen to rise much higher in the U.S. than in Scandinavia). This means that the surface of Earth is not flat, but curved, and the only model that fit observational data is that it it is (roughly) spherical. Q.E.D.