  Ly$\alpha$ forest absorption spectra decomposed into multiple Voigt profile
components (clouds) allow us to study clustering of intergalactic medium (IGM)
as a function of HI column density ($N_{\rm HI}$). Here, we explore the
transverse three-point correlation ($\zeta$) of these Ly$\alpha$ clouds using
mock triplet spectra obtained from hydrodynamical simulations at $z \sim 2$ on
scales of 1-5 $h^{-1}$cMpc. We find $\zeta$ to depend strongly on $N_{\rm HI}$
and scale and weakly on angle ($\theta$) of the triplet configuration. We show
that the "hierarchical ansatz" is applicable for scales $\ge~ 3h^{-1}$cMpc, and
obtain a median reduced three-point correlation (Q) in the range 0.2-0.7. We
show, $\zeta$ is influenced strongly by the thermal and ionization state of the
gas. As found in the case of galaxies, the influence of physical parameters on
Q is weaker compared to that of $\zeta$. We show difference in $\zeta$ and Q
between different simulations are minimized if we use appropriate $N_{\rm HI}$
cut-offs corresponding to a given baryon over-density ($\Delta$) using the
measured $N_{\rm HI}~vs~\Delta$ relationship obtained from individual
simulations. Additionally, we see the effect of pressure broadening on $\zeta$
in a model with artificially boosted heating rates. However, for models with
realistic thermal and ionization histories the effect of pressure broadening on
$\zeta$ is weak and sub-dominant compared to other local effects. We find
strong redshift evolution shown by $\zeta$, mainly originating from the
redshift evolution of thermal and ionization state of the IGM. We discuss the
observational requirements for the detection of three-point correlation,
specifically, in small intervals of configuration parameters and redshift.
