Existing models of galaxy formation have not yet been shown to explain key correlations between structure and star-formation activity. We offer a schematic model for quenching central galaxies that explains the tilted boundaries between star-forming and quiescent galaxies seen at all redshifts $z$ = 0-3 in their size$-$stellar mass and central-density$-$stellar mass distributions. The key idea is that larger star-forming galaxies (at fixed stellar mass) have lower black-hole masses owing to their lower central densities. Galaxies quench when the total effective energy radiated by their black holes equals an empirical function of their halo mass, and this function proves to be closely equal to a multiple of halo-gas binding energy. Since larger-radii galaxies start with smaller black holes, they must evolve to higher stellar masses before quenching, which produces the observed tilted boundaries. The quenching boundary proves to be almost parallel to the evolutionary tracks of individual galaxies; it is the evolution of the boundary, rather than the evolution of a galaxy, that leads to quenching. A candidate explanation for radii differences is differences in halo concentration and/or mean halo formation time at fixed halo mass. Implications include: radius is an important second parameter in shaping the life histories of galaxies and their black holes; most BH mass growth takes place in the green valley; galaxies enter the GV when their cumulative BH energy equals roughly 4$\times$ their halo-gas binding energy; star-forming, green valley, and quenched galaxies have different BH scaling laws; black holes are closely connected to their halos but in different ways for quenched, GV, and star-forming galaxies; and the same BH-quenching mechanism has existed since $z = 3$.