  The ordinary continued fractions expansion of a real number is based on the
Euclidean division. Variants of the latter yield variants of the former, all
encompassed by a more general Dynamical Systems framework. For all these
variants the Lagrange Theorem holds: a number has an eventually periodic
expansion if and only if it is a quadratic irrational. This fact is surely
known for specific expansions, but the only proof for the general case that I
could trace in the literature follows as an implicit corollary from much deeper
results by Boshernitzan and Carroll on interval exchange transformations. It
may then be useful to have at hand a simple and virtually computation-free
proof of a general Lagrange Theorem.
