Yes, I suppose the conspiracy theory mentality is susceptible to lots of different nonsense. What tickles me about the flat Earth theory is the number of conspiracy theories you have to believe in to make it work, i.e.

1. NASA existing (and being massively funded) primarily to perpetuate the spherical Earth hoax... for no clear benefit to anyone.
2. Believing that cold war era USA and USSR would actually collude on anything. Yet both created similar hoaxes about putting spacecraft into orbit.
3. Dismissing all photographic evidence as fake, regardless of its source.
4. Denying the existence of Australia (and presumably the rest of the Southern Hemisphere at some point) because the geography simply doesn't work for the Flat Earth model. Anyone who has been to these places was either tricked, or is part of the conspiracy.
5. Denying gravity, in favour of our disc constantly accelerating upwards, for no particular reason.
6. Denying the way that GPS and similar positioning systems work, without providing any rational alternative explanation. Clearly, anyone who's worked on any of these systems in any kind of depth is part of the conspiracy, too.

Anyone who will try that hard to hammer a square peg into a round hole, has the potential to believe anything. I wouldn't waste your breath trying to change his mind.

Someone like that is probably too far gone to be reasoned with sadly. If you do attempt it though, try asking cutting questions rather than giving proof. Questions can back him into a corner or force him to face contradictory beliefs.

Just take him to watch a sunset, as opposed to a sun moves away, and let him join the dots. If he doesn't get it don't waste your time

I wouldn't engage at all. I'm happy that they think the earth is flat.

Thanks for all the advice. I am leaning towards the idea of not engaging him on the topic unless he brings it up. If he does, I will listen to him and just ask him questions on his position and if possible why he holds them. I will not try to directly persuade him, because I think that will probably just get me worked up. I will probably just take it a step or two further than my sister-in-law did (just nodding and walking away from the discussion.)

The flat-earther stuff is stupid but basically harmless.

Alex Jones insistence that Sandy Hook and other mass shootings were faked is about as vile as you can get and still be classified as marginally human.

Some of those parents grieving those babies have received death threats.

Let me guess.

He denies the Holocaust too.

This comes to mind

It's probably fairly useless to try and convince a flat-earther that they're wrong... Just like it's useless to try and persuade a Christian to leave the faith.

If it were me, I'd just avoid discussion regarding the shape of the planet.

Here is a video about Engaging in Complex Conversations

I tend to ignore folks like that.

Here's an easy one. If you accucurately measure the angles in the triangle formed between Los Angeles, Atlanta, and Chicago, you with find they don't add up to 180. Why? Spherical Geometry. Here are two more. People in the northern hemisphere cannot see the Southern Cross. People in the southern hemisphere cannot see the North Star. If we lived in a flat Earth both of these constellations would be visible everywhere.

I heard about a long time before I finally decided to listen to what they have to say. Yes it is as dumb as I thought it was. lol.

Flat earthers have a hard time explaining seasons. I would start there and let him try to explain it. Tell him you will not debate him but want to hear him out with an open mind but he must state only facts and examples and, never use the words "I believe". Just listen intently. When you think, he's exhausted all his hypothesis on that subject, then ask him to explain a moon eclipse. Then a sun eclipse. Then tides. Then why is Mars round. Then how gravity works. By then he should be exhausted but you can finish by saying, how come in this age of social media and smart phones, nobody has ever traveled to the edge of earth and taken a picture and posted yet?

Relative to what?..

Dumb is the new cool.

No, I wouldn't engage him on any of his ridiculous beliefs. There's zero chance you can change his mind, no matter the evidence presented.

Voltaire said this: Those who can make you believe absurdities, can make you commit atrocities.

My favorite quote I saw by some high placed flat earth society member was something like, "There are 10s of 1,000s of us around the globe." ......idjits.