The basic issue here is mathematics. One way I did my calculations was using the 'law of cosines'. This law is interesting because it can be applied to Euclidean and non-Euclidean geometry. There are a few differences in the equations depending on whether you are applying it to a sphere or a flat plane. When I applied it to a sphere I did make the assumption that the sphere was perfectly round. The earth is a little elliptical and I didn't do any corrections for that but my answers should be accurate to within a couple hundreds of a percent.

When you look at the Wiki you only see a rough approximation of the radius of the earth so I wasn't expecting anything very accurate in the flat earth example. The main problem, I suspect, is the lack of mathematical skills of the average flat earth advocate. I was an engineer by training and took a whole lot of mathematics in college. It did take me quite a while to get up to speed on the mathematics of geodesy so I can imagine that it would be close to impossible for someone who only took a little math in high school to understand just how the equations really work. One the other hand, applying the law of cosines on a flat plane is a whole lot easier, but you will get a different answer if you apply it to a sphere.

There is really little serious hope for the flat earth model unless you can create a map with accurate distances AND bearings between all locations. If you study the math long enough you will eventually be convinced that the earth is a globe. I was also trained as a commercial pilot and a ship's officer. I know for a fact that the distances between airports and sea ports always are accurate. Otherwise there would be a potential safety issue. Additionally your GPS system is also based upon the earth being a sphere. Now I know that there will be some that will claim that the GPS system is based upon a bunch of transmitters on balloons. The crazy thing about that is it wouldn't matter. You could get a GPS system to work that way (but it wouldn't be practical). All the mathematics to calculate your position is all done in your receiver, and it's all based on the earth being a sphere. If you use the data transmitted from a GPS satellite and apply it to a flat plane you won't get a accurate answer. I've been on countless flights between the US, Asia, Europe, and the Middle East. It would be difficult for an airline to cheat on speeds and/or distances. Many people have GPS receivers that they take on airplanes. I'm one of them. My laptop had a very nice GPS receiver built in as well as the one on my iPhone. Both were known to be accurate because I compared them side by side with the GPS receivers on the ship that I worked on. Don't get the idea that shipping companies can cheat either. We often carried half a billion dollars worth of cargo and many shippers had their own GPS receiver coupled to a Sat C transmitter attached to the top of their container. They knew at all times just where their cargo was. There are multiple ship tracking websites so you can see the position, course & speed of any ship you are interested in. My wife often used one of these sites and could see just where I was in the world, in real time. Aircraft also have a satellite based system as well. If a customer has critical cargo aboard an international flight don't you think that someone in the office is keeping a close eye on what the plane is doing? Don't get the idea that either the distances between airports are inaccurate or that the airlines are cheating on speeds or distances. Sure, you can have either big head winds or tail winds that will effect the ground speed of an airliner. You can also have the captain intentionally slow the aircraft because of known traffic delays or a situation at the gate at the destination airport. I would say that more than 50% of the long flights I was on arrived either on time or ahead of schedule. Some were delayed because of bad weather.

So now the flat earth movement has another huge hurdle to get over. If you want to be taken seriously you have to have a map that is accurate and verifiable so airplanes and ships can accurately navigate between two points on the earth. Lives literally depend on it. Mine has been for years.