I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action.

I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbolic or elliptic geometry).

Searching on Google, I've found the article "Applying Burnside’s lemma to a one-dimensional Escher problem" by T. Pisanski, but it sounds to me rather a combinatorial result.