  We study the evolution of compact convex curves in two-dimensional space
forms. The normal speed is given by the difference of the weighted inverse
curvature with the support function, and in the case where the ambient space is
the Euclidean plane, is equivalent to the standard inverse curvature flow. We
prove that solutions exist for all time and converge exponentially fast in the
smooth topology to a standard round geodesic circle. This has a number of
consequences: first, to prove the isoperimetricinequality; second, to establish
a range of weighted geometric inequalities; and third, to give a counterexample
to the $n=2$ case of a conjecture of Gir\~ao-Pinheiro.
