Consider any triangle ABC. Connect the midpoints of each of the three sides. The inscribed triangle is equal to the other three triangles and they are all congruent. It turns out that the medians of the larger triangle are also the medians of the smaller, centrally inscribed, triangle.
(This is where it gets dicey)
I'd like to say that at this point, you could repeat this process on the smaller centrally inscribed triangle and then continue to do so infinitely. I'd then argue that at infinity, the points of the smallest triangle would be the same point. Tada, the medians are concurrent. Is this valid?
This was done for fun, I'd appreciate not being buried in a blizzard of Algebra or Calculus. Intuitively, does this work?