Curvature of Earth - Airplane
Nothing you say can convince this person that the Earth is round.  You need to show him.

Find a hill that is a few hundred feet high.  One of you stands at the top, the other at the bottom and observe the sunset.  You will, of course, be accompanied by a trusted flat-earth friend of his to validate your observations.

Each person will note the time that the sun disappears below the horizon.  Your times will be different.  You can even use math to calculate the expected time delta if you really want to drive your point home.  The higher the mountain, the greater the time delta.  Also, if you're at the top, you can flash him with a mirror to prove that you can still view the sun.  Be careful though...this might blow his mind.

This is a direct observation of the curvature of the Earth.  A rational person will accept this as proof of a curved surface.  An irrational bozo will continue to offer absurd theories.  Please let us know how your friend responds.