  It is shown that the effective interaction strength of three bosons at small
collision energies can be extracted from their wave function at zero energy. An
asymptotic expansion of this wave function at large interparticle distances is
derived, from which is defined a quantity $D$ named three-body scattering
hypervolume, which is an analog of the two-body scattering length. Given any
finite-range interaction potentials, one can thus predict the effective
three-body force from a numerical solution of the Schr\"{o}dinger equation. In
this way the constant $D$ for hard-sphere bosons is computed, leading to the
complete result for the ground state energy per particle of a dilute
Bose-Einstein condensate (BEC) of hard spheres to order $\rho^2$, where $\rho$
is the number density. Effects of $D$ are also demonstrated in the three-body
energy in a finite box of size $L$, which is expanded to the order $L^{-7}$,
and in the three-body scattering amplitude in vacuum. Another key prediction is
that there is a violation of the effective field theory (EFT) in the condensate
fraction in dilute BECs, caused by short-range physics. EFT predictions for the
ground state energy and few-body scattering amplitudes, however, are
corroborated.
