The chain of debate here goes like this:
1) We see published flight times and distances between distant cities. The agree with RET perfectly.
2) We look at the two FE maps published here and show that the distances shown on those maps are often 2 to 3 times LONGER than the airlines claim them to be.
3) The FE'ers claim that these times and distances are faked.
4) We can tell from "consumer watchdogs" (and government statistics, and the fact that there are some happy airline passengers) that these flights do in fact arrive on time - at least SOME of the time. If they make it on time even ONCE - then we know that the flight can be done - so the flight times that are published are not implausible.
5) We can look at the cruise speed of the airplanes they use and multiply that by the flight times (which we now know are plausible) to get a reasonable idea of what distance the planes are flying. This verifies that the distances quoted by the airlines are reasonable. Perhaps not spot on - but reasonable.
6) At this point, the FE'ers are in deep trouble. They know that if we RE'ers can provide measurements of distances then their pet theory is totally screwed.
7) But the only point of rational doubt is that the airlines literally don't know that the distances they publish are WILDLY incorrect...not off by 10% or 50% or even 100%. There are routes (Qantas' Sydney to Santiago Chile being a favorite here) that are over three times as long as the airlines "think" they are.
OK - so maybe the airlines ARE wrong! But we verified those distances against flight times that we KNOW must be correct. The only way that the airlines can have wrong distances is if their airplanes can fly two to three times faster than the airlines think they can!!!
9) But the cruise speed of a typical airliner is in the 500 to 600mph range - and three times that speed would be faster than Mach 2!! There is no possible way that something shaped like a 787 could fly that fast! Also, you'd hear sonic booms from airplanes that fly overhead! This isn't possible.
10) The other thing FE'ers will say is "Well, it's to do with the jet stream - it makes airplanes fly faster across the ground - so their airspeed can be subsonic - but their effective ground speed can be supersonic." - that's a reasonable guess.
11) But...when we look at these routes, the outbound and inbound flight times are within 15 to 20 minutes of each other - and the jet stream can't be blowing in both directions at once! And if it does blow East/West or West/East - then what about flight times for North/South and South/North flights? The jet stream idea is busted. The wind can't blow in all directions at once (and it's nothing like 1200mph - which is what it would need to be for the Qantas route to "work").
12) The FE'ers then (weakly, IMHO) flail around and come up with "Oh - but those maps you're taking distances from...those are just "examples" - we don't actually KNOW the true map of the Earth. (Darn! It would have been really nice if you'd mentioned this on your Wiki!).
13) So - suppose we take their (increasingly desperate) pleas to heart. Well how can we RE'ers argue that there isn't some "true" layout of the continents and oceans that would allow all of the flight distances to work out to be the same as they are on RE? We don't have that map - but how can we prove that there is no possible map?
14) Enter the "Quadrilateral Cities Proof" - this gets a bit mathematical - but it works out like this:
a) Pick four cities (A,B,C,D) that are widely spaced apart and which have excellent non-stop flights between them.
b) Draw a quadrilateral connecting them - and draw in the two diagonals of that quadrilateral.
b) Take the distances (after checking them against airplane cruise speeds and flight times) and fill in the lengths of those six lines.
c) Take three of the cities (A,B,C) and form a triangle out of them (BC is a diagonal of the quad) - and use those distances to calculate the angles between them using simple Euler math.
d) Take the cities (B,C,D) and form another triangle from them (BC still being the diagonal) - and using the fact that the four interior angles of a quadrilateral add up to 360 degrees, we know that fourth internal angle.
e) We now have enough information to calculate the length of that other diagonal (AD)...so let's do that.
f) Now we can compare the length of AD against the actual flight distance for AD...and...oh...there seems to be a big problem here. The real-world length of AD is MUCH longer than we calculated. We can do this again for the other diagonal, and the result is the same. Nothing works!
g) We can repeat this process for any number of cities and the answer is always the same. The length of the diagonals are always longer than they should be.
h) The REASON for this is that on a sphere, the lengths of diagonals of (not-flat) quadrilaterals are always longer than they would be on a flat plane.
i) HENCE there is no possible "Flat" map that can EVER match the real world flight distances.
15) The result of this is that there is no POSSIBLE flat earth map that can explain these airline flight distances. Even though we don't know how the map is laid out - we can use the "quadrilateral cities" proof to show that it cannot ever be possible for the Flat Earthers to construct a map that agrees with reality.
And remember - these errors are not small. Sure, there are winds and other airline realities - but they can't explain errors as big as three times what the "true" distance it.
QED.