One of my students created the following proof that the medians of a triangle are concurrent while waiting to talk to me in office hours.
Choose any two medians.
1) the two medians do meet somewhere inside the triangle.
2) If we join the centers of the three sides, obtaining a "central triangle",
then the two given medians are also medians of that central triangle,
hence their meeting point is also inside that triangle.
3) We are done.
I.e. iterating the procedure shows that any two medians meet at the unique point common to all central triangles.