  We study the synchronization of $N$ nearest neighbors coupled oscillators in
a ring. We derive an analytic form for the phase difference among neighboring
oscillators which shows the dependency on the periodic boundary conditions. At
synchronization, we find two distinct quantities which characterize four of the
oscillators, two pairs of nearest neighbors, which are at the border of the
clusters before total synchronization occurs. These oscillators are responsible
for the saddle node bifurcation, of which only two of them have a phase-lock of
phase difference equals $\pm$$\pi$/2. Using these properties we build a
technique based on geometric properties and numerical observations to arrive to
an exact analytic expression for the coupling strength at full synchronization
and determine the two oscillators that have a phase-lock condition of
$\pm$$\pi$/2.
