Simple check for the global shape of the Earth

I have been on a date recently, and everything went fine until the moment the girl has told me that the Earth is flat. After realizing she was not trolling me, and trying to provide her with a couple of suggestions why that may not be the case, I've faced arguments of the like "well, you have not been to the space yourself". That made me think of the following: I myself am certain that the Earth is ball-shaped, and I trust the school physics, but being a kind of a scientist, I could not help but agree with her that some of the arguments that I had in mind were taken by me for granted. Hence, I have asked myself - how can I prove to myself that the earth is indeed ball-shaped, as opposed to being a flat circle (around which the moon and the sun rotate in a convenient for this girl manner). Question: Ideally I want to have a proof that would not require travelling more than a couple of kilometers, but I am fine with using any convenient day (if e.g. we need to wait for some eclipse or a moon phase). For example, "jump an a plane and fly around the Earth" would not work for me, whereas "look at the moon what it is in phase X, and check the shape of the shade" would. Trick is, I know that it is rather easy to verify local curvature of the Earth by moving away from a tall object in the field/sitting on the beach and watching some big ship going to the horizon. However, to me that does not prove immediately that globally the Earth has same/similar curvature. For example, maybe it's just a shape of a hemisphere. So, I want to prove to myself that the Earth is ball-shaped globally, and I don't want to move much to do this. Help me, or tell me that this is not possible and why, please. As an example, most of the answers in this popular thread only focus on showing the local curvature. P.S. I think, asking how to use physics to derive global characteristics of an object from observing things only locally (with the help of the Sun and the Moon, of course) is a valid question,