  The Ly$\alpha$ forest decomposed into Voigt profile components (or "clouds")
allow us to study clustering properties of the intergalactic medium and its
dependence on various physical quantities. Here, we report the first detections
of longitudinal three-point ($\zeta$) and reduced three-point (Q) correlations
of low-$z$ (i.e $z<0.48$) Ly$\alpha$ clouds over a scale of $r_\parallel\leq 4$
pMpc. We measure $\zeta = 6.9\pm 1.3$ and Q = 1.7$\pm$0.4 at a transverse
separation of 1-2 pMpc. We show the amplitude of $\zeta$ increases with
increasing HI column density ($N_{\rm HI}$) while Q does not show any $N_{\rm
HI}$ dependence. We find the radial profile of $\zeta$ to be different for
high-$b$ (i.e $b>$40 km/s) and low-$b$ absorbers. About 84% of the triplets
contributing to $\zeta$ have nearby galaxies within a velocity separation of
500 km/s and a median impact parameter of 405 pkpc. The measured impact
parameters are consistent with appreciable number of triplets not originating
from individual galaxies but tracing the underlying galaxy distribution.
Frequency of occurrence of high-$b$ absorbers in triplets ($\sim$88%) is a
factor $\sim3$ higher than that found among the full sample ($\sim$32%). Using
four different cosmological simulations, we quantify the effect of peculiar
velocities, feedback effects and show that most of the observed trends can be
broadly reproduced. However, $\zeta$ at small scales ($r_\parallel<1$ pMpc) and
$b$-dependence of $\zeta$ in simulations are found inconsistent with the
observations. This could either be related to the fact that none of these
simulations reproduce the observed $b$-distribution and $N_{\rm HI}$
distribution for $N_{\rm HI}>10^{14}$ cm$^{-2}$ self-consistently or to the
widespread of signal-to-noise ratio in the observed data.
