Take the recent total eclipse as an example. It was predicted years (centuries) in advance using round earth mathematics. Is there even a single flat earther who was able to predict the eclipse using their model? If there is, I haven't heard.
That is probably the most profound rebuttal to flat-earthers I've heard. Granted, it's a grand extrapolation of "where do you think ships go when they sail over the horizon?", but it's an excellent summary.
I told her once that Alex Jones certainly has enough money to rent a jet to fly into a chemtrail and take some samples or take a ship and go straight and see what happens. But of course there tons of government conspiracies to prevent these things. In the end these people don't really want to know. They just have a deep discomfort with the current state of things.
This is a very succinct description of the mindset of the flat earthers I have talked with.
The outcome was mostly disappointment for me. They don't have anything like a reflectively consistent worldview. They don't in general have positive statements (other than that the earth is flat). Their beliefs are almost entirely negative, and phrased as arrogant dismissal of mainstream belief and evidence.
[1] a long long pendulum with a heavy weight
It is one thing to say that satellites and Foucault's pendulum prove that the earth is round, but if someone makes an argument (as in the linked article) that the visible peak of a mountain should be 900 meters below the horizon, and the round earthers response to that is some poorly exposited fact about refraction, it doesn't help the round Earth case in the eyes of the flat earther who raised that point.
A better response would be to include some references to some facts about atmospheric refraction in addition. Hopefully the flat earther might understand it.
So in effect I am not actually a flat Earther, but round earthers should be able to explain observations which are not consistent with round earth expectations better.
In my experience it doesn't make any difference how many references you include. They either don't read them at all, or they read them but don't address the content. So far I have been unable to get a flat earther to respond to basic information that I provide that contradicts their assertions.