Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then we have $$ \frac{l_X(c)}{K}\leq l_Y(\phi(c))\leq Kl_X(c)$$
For any simple closed curve $c$. Now as an immediate corollary on the Farb Margalit it is given the following.
Assume $d_T(X,Y)=d$, then for any $c$ as above we have $$ \frac{l_X(c)}{e^{2d}}\leq l_Y(c)\leq e^{2d}l_X(c)$$
The fact why the constant turns out to be the exponential is clear to me (def. Of K), I don't get why in the middle part we don't have $\phi(c)$ for $\phi$ this time a Teichmüller map according to my understandings.
Can we always assume that such map fixes a curve $c$? I dont think so, because this would be a contradiction to the classification of elements in $Mod(S)$ according to me.
So how do we get the central term?