  We aim to unravel the interplay between bars, star formation (SF), and active
galactic nuclei (AGNs) in barred galaxies. To this end, we utilize the SDSS
DR12 to select a sample of nearby (0.02 < z < 0.06) disk galaxies that are
suitable for bar examination ($M_r < -20.12$ and inclination $\lesssim$
53$^{\circ}$). We identify 3662 barred galaxies and measure the length and axis
ratio of each bar. We invent new bar parameters that mitigate the stellar and
bulge mass biases and show, for the first time, that the evolution of non-AGN
and AGN-hosting barred galaxies should be tracked using different bar
parameters; the bar length for non-AGN galaxies and the bar axis ratio for
AGN-hosting galaxies. Our analysis confirms that barred galaxies have a higher
specific SF rate than unbarred control galaxies. Moreover, we find a positive
correlation of bar length with both the SF enhancement and the centrally
star-forming galaxy fraction, indicating the interconnectivity of bars and SF
through the bar-driven gas inflow. We also find that while the AGN fraction of
barred galaxies is the same as that of the unbarred control sample, galaxies
hosting more massive black holes (BHs) have rounder (i.e., higher axis ratio)
bars, implying that the bar is not a cause of AGN activity; rather, AGNs appear
to regulate bars. Our findings corroborate theoretical predictions that bars in
non-AGN galaxies grow in length, and bars in AGN-hosting galaxies become
rounder as BHs grow and eventually get destroyed.
