Finding a curve orthogonally equidistant between two curves.

Question

I have two curves, $f_1(x)$ and $f_2(x)$ which meet at $(x_0, 0)$ and $(x_1, 0)$, these curves as such form a closed shape, but not necessarily a convex one. I would like to generate a curve $f_c(x)$ between the two whose (euclidean) distance orthogonal to the curve itself is equal between the two outer curves. Clearly the average of the two functions is not the answer. It seems to me the best guess is a 2nd order ODE, 2 point BVP, similar to a strom-liouville problem, based on the evolution of the $f_c(x)$ beginning at $(x_0, 0)$, but I am unsure of the exact expression of the ODE.