take a cyclic quadrilateral. there are two ways to divide it into two triangles, using a diagonal. prove that the sum of the two inradii of the triangles is the same, no matter what your choice of diagonal is. (you can easily generalize this to any cyclic polygon divided into triangles.)
it's such a simple fact, i can't believe i haven't seen it in any exposition of theorems. but i've only seen complex proofs of it. i hope it generalizes into some big theory, like inversion. that would be exciting.
i also wonder if the converse is true... if the sums of the two inradii are equal, is the quadrilateral cyclic necessarily?