  The {\it Linear Point} (LP), defined as the midpoint between the BAO peak and
the associated left dip of the two-point correlation function (2PCF), $\xi(s)$,
is proposed as a new standard ruler which is insensitive to nonlinear effects.
In this paper, we use a Bayesian sampler to measure the LP and estimate the
corresponding statistical uncertainty, and then perform cosmological parameter
constraints with LP measurements. Using the Patchy mock catalogues, we find
that the measured LPs are consistent with theoretical predictions at 0.6 per
cent level. We find constraints with midpoints identified from the rescaled
2PCF ($s^2 \xi$) more robust than those from the traditional LP based on $\xi$,
as the BAO peak is not always prominent when scanning the cosmological
parameter space, with the cost of 2--4 per cent increase of statistical
uncertainty. This problem can also be solved by an additional dataset that
provides strong parameter constraints. Measuring LP from the reconstructed data
slightly increases the systematic error but significantly reduces the
statistical error, resulting in more accurate measurements. The 1$\,\sigma$
confidence interval of distance scale constraints from LP measurements are
20--30 per cent larger than those of the corresponding BAO measurements. For
the reconstructed SDSS DR12 data, the constraints on $H_0$ and $\Omega_{\rm m}$
in a flat-$\Lambda$CDM framework with the LP are generally consistent with
those from BAO. When combined with Planck cosmic microwave background data, we
obtain $H_0=68.02_{-0.37}^{+0.36}$ ${\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ and
$\Omega_{\rm m}=0.3055_{-0.0048}^{+0.0049}$ with the LP.
