^^^ Just kill yourself now:
Mathematical proof that modifying the experiment by adding a 3rd well (or stick) at Luxor doesn't change anything.
Modern science places the earth’s radius at 3,959 miles, which gives a pole to pole circumference of about 24,875 miles. Dividing by 360, we find that one degree of arc should be about 69.1 miles on the earth’s surface. That is why lines of latitude are about 69.1 apart. So on a globe for every 69.1 miles or one degree of latitude you move north or south of the point the sun is directly overhead, the angle of the shadow in that location would increase by one degree. If Eratosthenes was correct, the angle of his shadow must have been equal to Alexandria’s latitude north of Syene, because that would be the angle that Alexandria is leaning away from Syene due to the curvature of the earth. If three other astronomers had performed the experiment on the same day at 20 degrees, 40 degrees, and 60 degrees north of the Tropic of Cancer, they would each respectively measure a shadow angle of 20 degrees, 40 degrees, and 60 degrees, and all would arrive at a circumference of 24,875 miles. The beauty of Eratosthenes’ experiment is that you can repeat it anywhere on earth, on any day of the year, and arrive at the same circumference. This is where the flat earth model runs into problems.
In the flat earth model, as a consequence of geometry, as your distance from the point the sun is directly overhead increases, the distance between each successive degree of shadow will also increase exponentially. For example, assume the sun is giving off a 45 degree shadow at 45 degrees north, as depicted in this flat earther meme. If the earth were flat, that would mean that the sun is, more precisely, 3,110 miles high. In that case, it would be 1,132 miles to the 20 degree shadow, 1,478 miles between the 20 and 40 degree shadows, and a whopping 2,777 miles between the 40 and 60 degree shadows. Someone performing Eratosthenes’ experiment at 20 degrees north of the Tropic of Cancer would arrive at a circumference of 20,376 miles, someone at 40 degrees north would get 23,486 miles, and someone at 60 degrees north would get 32,320 miles.
Flat earthers use the Azimuthal Equidistant Projection map, which shows all points at an undistorted distance and direction from the center. Lines of latitude on the flat earth map have flat earth mapthe same spacing as those on the globe: 69.1 miles per degree. It is for this reason that the angle of the shadow can only be equal to the degrees of latitude between you and the point the sun is directly overhead in one location. Everywhere south of that point the shadow angle would be greater than the latitude and everywhere north of that point the shadow angle would be less than the latitude. If the sun were 3,110 miles above the flat earth, the 20 degree shadow would be 16.4 degrees north, the 40 degree shadow 37.8 degrees north, and the 60 degree shadow 78 degrees north. Only at 45 degrees north would the latitude and the angle of the shadow actually match as they would everywhere on the globe and only there would someone arrive at a circumference of 24,875 miles. You can change the height of the sun all you like, the latitude and angle will only be the same in one location. Even then, it is only a coincidence due to the random height of the sun, not a direct function of latitude as it is on the globe.
To be sure, if each of our astronomers were to measure a shadow angle corresponding to their latitude, as is to be expected on the globe, and you were still assuming a flat earth model, that would require that from 20 degrees north the sun be 3,797 miles high above the flat earth, at 40 degrees north 3,294 miles high, and at 60 degrees north 2,393 miles high. An impossibility. The only possible solution would be to distort the flat earth map beyond recognition by staggering the lines of latitude, but even then it would only be correct for a single day, as the sun moves to a different latitude each day and the model would again become increasingly wrong.
