  This work reports on a new reconstruction algorithm allowing to reconstruct
the decays of ortho-positronium atoms into three photons using the places and
times of photons recorded in the detector. The method is based on trilateration
and allows for a simultaneous reconstruction of both location and time of the
decay. Results of resolution tests of the new reconstruction in the J-PET
detector based on Monte Carlo simulations are presented, which yield a spatial
resolution at the level of 2 cm (FWHM) for X and Y and at the level of 1 cm
(FWHM) for Z available with the present resolution of J-PET after application
of a kinematic fit. Prospects of employment of this method for studying angular
correlations of photons in decays of polarized ortho-positronia for the needs
of tests of CP and CPT discrete symmetries are also discussed. The new
reconstruction method allows for discrimination of background from random
three-photon coincidences as well as for application of a novel method for
determination of the linear polarization of ortho-positronium atoms, which is
also introduced in this work.
