Something that I've just started doing is enhancing my answer to a student's question by sometimes formulating (when possible, of course) a theorem and proof that are directly related to the question. For example, when one of my students asked me to describe the symmetry of an odd function, i.e., a function for which f(x)= -f(-x), the theorem I cooked up was:
For an odd function:
a) The line segment joining f(x) and f(-x) passes through the origin, and
b) The origin bisects that line segment
Even though the homework problem that a student does not request such a theorem, posing it and proving it enhances the student's understanding of the concept and makes it more interesting.