  The formation of close binaries has been an open question for decades. A
large fraction of close binaries are in triple systems, suggesting that their
formation may be associated with the Kozai-Lidov mechanism. However, this
picture remains under debate because the configurations of many observed
triples are unlikely to trigger the Kozai-Lidov mechanism. In this paper, we
use the close binary samples, including eclipsing, spectroscopic, and
astrometric binaries, from Gaia Data Release 3 to investigate the mysterious
connection between inner binaries and their wide tertiaries. We show that the
wide tertiary (at $10^3$-$10^4$ AU) fraction increases with decreasing orbital
periods of the inner binaries. The wide tertiary fraction of eclipsing binaries
(a median orbital period of $0.41$ day) is $2.33\pm0.11$ times higher than the
field wide binary fraction. Furthermore, there is a tentative excess at
$\sim10^4$ AU for tertiaries of eclipsing binaries. Using the $v$-$r$ angle
distributions, we show that wide tertiaries have isotropic orientations with
respect to the inner binaries, and the co-planar orbits can be ruled out. The
inferred eccentricity distribution of wide tertiaries is consistent
($<1\sigma$) with being thermal ($f(e)\propto e$), similar to wide binaries at
similar separations. The dynamical unfolding scenario is disfavored because it
predicts highly eccentric wide tertiaries, which is inconsistent with our
findings. For the Kozai-Lidov mechanism to be effective for wide tertiaries at
$>10^3$ AU, the initial separations of the inner binaries need to be $>3$ AU.
Future theoretical investigations are needed to explore the parameter space at
these large initial separations and large tertiary separations.
