  Geometric distortion (GD) critically constrains the precision of astrometry.
Its correction required GD calibration observations, which can only be obtained
using a special dithering strategy during the observation period. Some
telescopes lack GD calibration observations, making it impossible to accurately
determine the GD effect using well-established methods. This limits the value
of the telescope observations in certain astrometric scenarios, such as using
historical observations of moving targets in the solar system to improve their
orbit. We investigated a method for handling GD that does not rely on the
calibration observations. With this advantage, it can be used to solve the GD
models of telescopes which were intractable in the past. Consequently,
astrometric results of the historical observations obtained from these
telescopes can be significantly improved. We used the weighted average of the
plate constants to derive the GD model. The method was implemented in Python
and released on GitHub. It was then applied to solve GD in the observations
taken with the 1-m and 2.4-m telescopes at Yunnan Observatory. The resulting GD
models were compared with those obtained using well-established methods to
demonstrate the accuracy. Furthermore, the method was applied in the reduction
of observations for two targets, the moon of Jupiter (Himalia) and the binary
GSC2038-0293, to show its effectiveness. The GD of the 60-cm telescope at
Yunnan Observatory without calibration observations was accurately solved by
our method. After GD correction, the astrometric results for both targets show
improvements. Notably, the mean residual between observed and computed position
($O-C$) for the binary GSC2038-0293 decreased from 36 mas to 5 mas.
