  The HST/ACS colour-magnitude diagrams (CMD) of the populous LMC star cluster
NGC1751 present both a broad main sequence turn-off and a dual clump of red
giants. We show that the latter feature is real and associate it to the first
appearance of electron-degeneracy in the H-exhausted cores of the cluster
stars. We then apply to the NGC1751 data the classical method of star formation
history (SFH) recovery via CMD reconstruction, for different radii
corresponding to the cluster centre, the cluster outskirts, and the underlying
LMC field. The mean SFH derived from the LMC field is taken into account during
the stage of SFH-recovery in the cluster regions, in a novel approach which is
shown to significantly improve the quality of the SFH results. For the cluster
centre, we find a best-fitting solution corresponding to prolonged star
formation for a for a timespan of 460 Myr, instead of the two peaks separated
by 200 Myr favoured by a previous work based on isochrone fitting. Remarkably,
our global best-fitting solution provides an excellent fit to the data - with
chi^2 and residuals close to the theoretical minimum - reproducing all the CMD
features including the dual red clump. The results for a larger ring region
around the centre indicate even longer star formation, but in this case the
results are of lower quality, probably because of the differential extinction
detected in the area. Therefore, the presence of age gradients in NGC1751 could
not be probed. Together with our previous findings for the SMC cluster NGC419,
the present results for the NGC1751 centre argue in favour of multiple star
formation episodes (or continued star formation) being at the origin of the
multiple main sequence turn-offs in Magellanic Cloud clusters with ages around
1.5 Gyr.
