So, TFES' counter-argument for the Quadrilateral Proof is that we don't know if the distances are correct. All right.

What if we prove that for any FE map configuration that

*does* get the angles right by using other distances, therefore avoiding the proof, those other distances will

*still not match up* with the flight times and speeds?

If the Earth is flat, there MUST be a constellation where BOTH

1. The Quadrilateral Proof gives

**identical angles**2. The times that it would take planes to fly the distances

**matches those given by airlines** like Qantas, calculatable by dividing the distances by the cruise speed of the planes used / Alternatively: The cruising speed the plane would need to have in order to complete its journey in the given time matches the one that is given for each plane type and flight

3. (optional) The angles between the cities would have to remain roughly the same (not sure, since FET seems to question those angles anyway)

If there is no such constellation,

**then the Earth cannot be flat**. Shouldn't be too hard.

I will be using

http://www.calculator.net/triangle-calculator.html for the angles.

Flight distances will be taken from

WorldAtlas (all in km). The corner points are New York (NY), Paris (PA), Buenos Aires (BA) and Cape Town (CT).

NY - PA 5919

NY - BA 8383

NY - CT 12472

CT - PA 9148

CT - BA 6938

BA - PA 10930

Result for angle at NY calculated directly: 98.201°

Result for angle at NY calculated by adding: 31.814° + 43.611° = 76° => Difference of ~22°

Okay, so as we all can see, with the data we get from WorldAtlas, Earth cannot be flat. So let’s just change those numbers up a bit. At this point I’m not sure yet, but I think in order for the second angle to get closer to the first one, we’ll have to make NY – CT and BA – PA a bit shorter. Shall we? Let's say NY – CT 10000; BA – PA 8000, the other distance figures stay the same.

Result for angle at NY calculated directly: 65.401°

Result for angle at NY calculated by adding: 43.24° + 64.294° = 108° => Difference of ~43° in the other direction

Okay, seems like I went a little bit overboard, but the angles did go into the direction I wanted them to. We can change the relevant angles by changing those two distances. Let’s try NY – CT 11000 and BA – PA 9500. The rest stays the same.

Result for angle at NY calculated directly: 81.272°

Result for angle at NY calculated by adding: 56.248 + 39.092° = 95°

We’re getting closer to distances that would work on a FE, but NY – CT and PA – BA still seem to be a little too short. Considering the huge effect it had before, let’s just add 500 km to each figure. NY – CT 11500 and BA – PA 10000.

Result for angle at NY calculated directly: 86.993°

Result for angle at NY calculated by adding: 36.8° + 52.115° = 88.915°

We’re getting really close now – so close that the digits after the dot will soon get important, so I stopped rounding the sums out of laziness. Our added angle is two degrees greater than the first one. I’m going to add a smaller bit of length: NY – CT 11550 and BA – PA 10050.

Result for angle at NY calculated directly: 87.513°

Result for angle at NY calculated by adding: 36.561° + 51.695° = 88.256°

The two angles are now less than a degree apart – a mere 0.743°. I’ll try adding just 30 km more to both distances. NY – CT 11580 and BA – PA 10080.

Result for angle at NY calculated directly: 87.862°

Result for angle at NY calculated directly: 36.417° + 51.442° = 87.859°

(I have proof for those angles and distances -

87.862°,

36.417° and

51.442°)

As you can see, the difference between the angle calculated from NY – PA, NY – BA and BA – PA is now only 0.003 degrees smaller than the one added up using the other distances. If the distances were like this in the real world, then mathematically the world could be flat.

Summing the first part up, our distances could be:

NY - PA 5919

NY - BA 8383

NY - CT 11580 – now 892 km less than before

CT - PA 9148

CT - BA 6938

BA - PA 10080 – now 850 km less than before

Note that we could also have done this by increasing the other distances, instead of decreasing those two – or a combination of both, which would probably give more accurate results, but I don’t think it matters. If anyone wants to put in the extra work (I’m looking at you, Flat Earthers –

*this might actually be a method to create a more or less accurate flat map!*), you’re very much welcome to do that. I'll edit my post accordingly.

Now, before I start calculating with the flight times and velocities of the planes, does anyone have objections to how I'm doing this? Because I don’t want to do all that work without making sure it doesn’t get discredited again.