Reasons Why Nearby Sun Is Impossible
Parallax
A nearby Sun would produce a profound amount of parallax motion that would be obvious to a moving observer or observers in different locations.
Parallax motion is like when you drive down the road and the nearby trees appear to move through your field of view very rapidly, slightly more distant objects move more slowly, distant clouds move even more slowly, and a sun that is only some 3000 miles away would move as well.
But we observe the sun remaining in the same place even when viewed from fairly distant locations at the same time.
The amount of angular difference we should see in a 3000 mile distant sun for even just 100 miles between observers would be 1.9°! The sun is only 0.5° across in the sky so this would be almost FOUR solar distances shifted. This would be trivial to observe but in actuality the angle difference is too small to be measured without extremely specialized and costly equipment (less than one quarter of an arcsecond for observers 100 miles apart).
As you move further apart you also move around the curvature of the Earth so you start to measure the angle of curvature in addition to the angle to the Sun, we deal with this in Height of Sun.
Speed of Sun
With a nearby Sun it would also be impossible to maintain the same apparent motion of the Sun in winter time as in summer because the Tropic of Capricorn would be a much greater distance around than the Tropic of Cancer. So the Sun would either have to go a lot faster or it would take a lot longer to go all the way around.
Let's see  the Equator is 24,901 miles around that would mean a Flat Earth distance to the North Pole of 24,901/(2π) = 3963.12 miles  since 22.5° is 1553 miles further that gives us a circumference of 5516.12(2π) = 34658.8 miles.
So the sun would have to go 39% faster to keep the length of a day the same but #1 days stay 24 hours year around and #2 nobody notices the Sun racing overhead in the winter. So this Flat Earth model is impossible.
Height of Sun
From a right triangle we can calculate the height a if we have the distance b and angle α using the equation:
a = b × tan(α)
Some people get confused by the tangent function but this function is nothing but SIMPLE division because tan(α) means nothing more than taking the side opposite our angle α  which would be side a, divided by the side that is adjacent to our angle α  which is side b  so it's nothing more than tan(α) = a / b. You might also recognize (a / b) as being RISE/RUN, or the slope of a line and this is exactly what it is!
So this means these are all saying exactly the same thing:
a = b × SLOPE
a = b × tan(α)
a = b × (a/b)
This isn't complex or scary math. It just means that the angle and the slope are really the same thing and we can convert between them. They are just expressed in different units.
From this you can also see why when a and b are equal in length then a/b would be 1 and sure enough, tan(45°) = 1  this value 1 is the slope of the line. So 45° just means the slope is 1.
IF the Earth was flat then we could use this formula to find the height of the Sun!
Street Lamp Example
Let's take a simple example first:
Standing at point C (from our diagram above) a street lamp is directly overhead. We walk 10 meters to point A and find that our angle α is 45°  10 meters * tan(45°) = 10 meters so our light is 10 meters up (remember tan(45°) = 1 so this is trivial). Easy enough. Now let's walk out to 20 meters away from C. Since we doubled our distance that means our slope is onehalf:
Now we find the height of the lamp is 20 * 0.5  this formula still gives us 10 meters. We didn't even need to use trigonometry because we already know the slope of the line  but we find the angle is 26.5650512°
No matter what distance you pick you can find the SAME height by multiplying in your distance times the slope of the line (or finding the slope from the angle using tan(α)).
Does This Work on Earth?
The question now is, does this actually work when we measure the Sun from Earth? If the Earth is curved then our segment b is curved and this function will NOT give a consistent value for the height of the Sun because the angles will be off because our horizon would be rotating along with us as we moved further North, like so:
In this case our 45° is REALLY 90° because our horizon is tilted by about 45°
So let's take some measurements at different latitudes and try to calculate the height of the Sun from those.
If we look at data from the 2016 Vernal Equinox which occurred at Sun, 20 Mar 2016 04:29:48 UTC We find that the longitude where the sun is directly overhead is at Longitude: 114° 24' 6.8394" (or 114.4019 in decimal form). From here we can find the angle of the sun from the horizon at various latitudes, and we note that every 10° latitude is about 1111 km. If you prefer to use more exact values you can find them using the Great Circle distance calculator, I recommend the WGS84 data as the most accurate. The distance between latitude lines is not exactly equal because the Earth is a very slightly oblate spheroid and not a perfect sphere. However, the differences are small enough that they don't really matter for what we are doing here.
Here is our data... isn't it funny how the angle of the sun is right at 90° minus our latitude? This would NOT be the case on a Flat Earth. If the Earth was flat the angles would be slowly approaching 26.565° (from our streetlight example we know that if we double the distance it should make the slope one half) rather than approaching 0° and we would get a constant Height back as we did with the streetlamp.
Reminder, our height column here is given by our formula: a = b × tan(α)
Latitude  Observed Angle Of Sun At Equinox 
Distance from Equator (miles) 
Which Height Is It? 

10°

80°

690.3  3915.1 
20°

70.01°

1380.0  3793.6 
30°

60.01°

2070.3  3587.4 
40°

50.01°

2760.7  3291.2 
45°

45.02°

3105.2  3107.3 
50°

40.02°

3450.3  2897.2 
60°

30.03°

4140.0  2393.1 
70°

20.04°

4829.6  1761.7 
80°

10.09°

5516.5  981.7 
89°

1.36°

6119.2  145.3 
But with the sun angles we observe the Flat Earth model clearly does not work.
It is Impossible for the Sun to be at 3915.1 miles and 145.3 miles up at the same time and those values are so far apart it cannot be simple measurement error. It is just fundamentally wrong.
The Earth is curved.
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