  We consider a model with $\rho\pi\pi$, $\rho KK$, and gauge-invariant
$\rho\gamma$ couplings and obtain the pion form factor $F_\pi$ at timelike
momentum transfers by resummation of pion and kaon loops. We use a dispersion
representation for the loop diagrams and analyse ambiguities related to
subtraction constants. The resulting representation for $F_\pi$ is shown to
have the form of the conventional vector meson dominance formula with one
important distinction - the effective $\rho$-meson decay constant $f^{\rm
eff}_\rho$ turns out to depend on the momentum transfer. For the
electromagnetic pion form factor we include in addition the $\rho-\omega$
mixing effects. We apply the representations obtained to the analysis of the
data on the pion form factors from $e^+e^-$ annihilation and $\tau$ decay and
extract the $\rho^-$, $\rho^0$ and $\omega$ masses and coupling constants.
