Paul Carter, David Lowry-Duda, On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions, The American Mathematical Monthly, Vol. 124, No. 6 (June-July 2017), pp.

Then at each stage the midpoints of these corners and the centre point are displaced by an amount = where is the average value of the adjacent corners, decreases at each stage (say, by a factor of 2) ...

I'll let you work out your own proof of this theorem, but I'll give you a hint. Here's a useful fact: in any triangle, if you connect the midpoints of two sides by a line segment, then that ...