Background
Given a manifold $M$ and a smooth curve ${C}$ between $p,q \in M$ given by $c: [a,b] \to M$, we want to construct the properties for a generic candidate function $F:TM \to \mathbb{R}_{+}$ which induces a length $$ L_{F}(C):= \int_{a}^b F(c(t),c'(t)) dt. $$
and a distance function $$ d_F(p,q) := \displaystyle\mathrm{inf}_{C} \ L_{F}(C), \quad \text{where} \ C \ \text{is a smooth curve connecting } p \ \text{and } q.$$
One desirable property would be that $\mathbf{F}$ determines $\mathbf{d_F}$ uniquely. The book I am reading seems to suggest that it is enough to assume the triangle inequality for $F$ to get this property. However, for some reason I am finding it difficult to work out the implication, and would appreciate strong initial pointers.
Question
In other words, we are trying to show that, under the triangle inequality assumption for $F$ and $G$, we have $$d_F(p,q) = d_G(p,q) \ \ \forall p,q \in M \implies G=F.$$
I've tried very basic stuff not worth writing down but cannot seem to find the basic idea of a proof, if any, so any pointers are welcome.