  Alkali trimers, Ak$_3$, located on the surface of He nanodroplets are triply
ionized following multiphoton absorption from an intense femtosecond laser
pulse leading to fragmentation into three correlated Ak$^+$ ions. Combining the
information from three-fold covariance analysis of the emission direction of
the fragment ions and from their kinetic energy distributions
$P(E_{\text{kin}})$, we find that Na$_3$, K$_3$, and Rb$_3$ have an equilateral
triangular structure, corresponding to that of the lowest-lying quartet state
$^{4}\mathrm{A}_{2}'$, and determine the equilibrium bond distance
$R_\text{eq}$(Na$_3$) = 4.65 $\pm$ 0.15 {\AA}, $R_\text{eq}$(K$_3$) = 5.03
$\pm$ 0.18 {\AA}, and $R_\text{eq}$(Rb$_3$) = 5.45 $\pm$ 0.22 {\AA}. For K$_3$
and Rb$_3$ these values agree well with existing theoretical calculations,
while for Na$_3$ the value is 0.2-0.3 {\AA} larger than the existing
theoretical results. The discrepancy is ascribed to a minor internuclear motion
of Na$_3$ during the ionization process. Also, we determine the distribution of
internuclear distances $P(R)$ under the assumption of fixed bond angles. The
results are compared to the square of the internuclear wave function
$|\Psi(R)|^2$.
