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.
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