  We obtain an exact result for the midpoint probability distribution function
(pdf) of the stationary continuum directed polymer, when averaged over the
disorder. It is obtained by relating that pdf to the linear response of the
stochastic Burgers field to some perturbation. From the symmetries of the
stochastic Burgers equation we derive a fluctuation-dissipation relation so
that the pdf gets given by the stationary two space-time points correlation
function of the Burgers field. An analytical expression for the latter was
obtained by Imamura and Sasamoto [2013], thereby rendering our result explicit.
In the large length limit that implies that the pdf is nothing but the scaling
function $f_{{\rm KPZ}}(y)$ introduced by Pr\"ahofer and Spohn [2004]. Using
the KPZ-universality paradigm, we find that this function can therefore also be
interpreted as the pdf of the position y of the maximum of the Airy process
minus a parabola and a two-sided Brownian motion. We provide a direct numerical
test of the result through simulations of the Log-Gamma polymer.
