  Consider the barycentric subdivision which cuts a given triangle along its
medians to produce six new triangles. Uniformly choosing one of them and
iterating this procedure gives rise to a Markov chain. We show that almost
surely, the triangles forming this chain become flatter and flatter in the
sense that their isoperimetric values goes to infinity with time. Nevertheless,
if the triangles are renormalized through a similitude to have their longest
edge equal to $[0,1]\subset\CC$ (with 0 also adjacent to the shortest edge),
their aspect does not converge and we identify the limit set of the opposite
vertex with the segment [0,1/2]. In addition we prove that the largest angle
converges to $\pi$ in probability. Our approach is probabilistic and these
results are deduced from the investigation of a limit iterated random function
Markov chain living on the segment [0,1/2]. The stationary distribution of this
limit chain is particularly important in our study. In an appendix we present
related numerical simulations (not included in the version submitted for
publication).
