The Linear Point (LP), defined as the mid-point between the baryon acoustic oscillation (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 non-linear 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 mid-points identified from the rescaled 2PCF (s(2)xi) more robust than those from the traditional LP based on., 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 data set 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 sconfidence interval of distance scale constraints from LP measurements are 20-30 per cent larger than those of the corresponding BAO measurements. For the reconstructed Sloan Digital Sky Survey Data Release 12 data, the constraints on H-0 and Omega(m) in a flat-Lambda cold dark matter 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) km s(-1) Mpc(-1) and Omega(m) = 0.3055(+-0.0048)(-0.0049) with the LP.