  We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de
Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the
inscribed angle theorem, the law of sines, the circumradius, inradius and some
angle bisector formulae, by assuming the existence of an unknown relation
between the geometric quantities at stake, observing how the relation behaves
under small deviations of those quantities, and naturally establishing
differential equations that we integrate out. Applying the general solution to
some specific situation gives a particular solution corresponding to the
expected theorem. We also establish an equivalence between a polynomial
equation and a set of partial differential equations. We finally comment on a
differential equation which arises after a small scale transformation and
should concern all relations between metric quantities.
