  We show how to extract a monotonic learning algorithm from a classical proof
of a geometric statement by interpreting the proof by means of interactive
realizability, a realizability sematics for classical logic.
  The statement is about the existence of a convex angle including a finite
collections of points in the real plane and it is related to the existence of a
convex hull. We define real numbers as Cauchy sequences of rational numbers,
therefore equality and ordering are not decidable. While the proof looks
superficially constructive, it employs classical reasoning to handle
undecidable comparisons between real numbers, making the underlying algorithm
non-effective.
  The interactive realizability interpretation transforms the non-effective
linear algorithm described by the proof into an effective one that uses
backtracking to learn from its mistakes. The effective algorithm exhibits a
"smart" behavior, performing comparisons only up to the precision required to
prove the final statement. This behavior is not explicitly planned but arises
from the interactive interpretation of comparisons between Cauchy sequences.
