This is one way:
FullSimplify[Solve[(x - x2)^2 + (f1 - y2)^2 == L2^2, x],
{x1 == x3 + x2, y1 == y2 + y3}]
The assumptions are added as second argument for Simplify or FullSimplify.
Another way can be to use replacement rules:
FullSimplify[Solve[(x - x2)^2 + (f1 - y2)^2 == L2^2, x]] /.
{x1 -> x3 + x2, y1 -> y2 + y3}
