We report the discovery of three extrasolar planets that transit their moderately bright (m V= 12-13) host stars. WASP-44b is a 0.89-M Jup planet in a 2.42-day orbit around a G8V star. WASP-45b is a 1.03-M Jup planet which passes in front of the limb of its K2V host star every 3.13 days. Weak Ca ii H&K emission seen in the spectra of WASP-45 suggests that the star is chromospherically active. WASP-46b is a 2.10-M Jup planet in a 1.43-day orbit around a G6V star. Rotational modulation of the light curves of WASP-46 and weak Ca ii H&K emission in its spectra show the star to be photospherically and chromospherically active. We imposed circular orbits in our analyses as the radial-velocity data are consistent with (near-)circular orbits, as could be expected from both empirical and tidal-theory perspectives for such short-period, ∼Jupiter-mass planets. We discuss the impact of fitting for eccentric orbits for such planets when not supported by the data. The derived planetary and stellar radii depend on the fitted eccentricity and these parameters inform intense theoretical efforts concerning tidal circularization and heating, bulk planetary composition and the observed systematic errors in planetary and stellar radii. As such, we recommend exercising caution in fitting the orbits of short-period, ∼Jupiter-mass planets with an eccentric model when there is no evidence of non-circularity.