Much of the structure in social networks has been explained by two seemingly independent network evolution mechanisms: triadic closure and homophily. While it is common to consider these mechanisms separately or in the frame of a static model, empirical studies suggest that their dynamic interplay is the very process responsible for the homophilous patterns of association seen in off- and online social networks. By combining these two mechanisms in a minimal solvable dynamic model, we confirm theoretically the long-held and empirically established hypothesis that homophily can be amplified by the triadic closure mechanism. This research approach allows us to estimate how much of the observed homophily in various friendship and communication networks is due to amplification for a given amount of triadic closure. We find that the cumulative advantage-like process leading to homophily amplification can, under certain circumstances, also lead to the widely documented core-periphery structure of social networks, as well as to the emergence of memory of previous homophilic constraints (equivalent to hysteresis phenomena in physics). The theoretical understanding provided by our results highlights the importance of early intervention in managing at the societal level the most adverse effects of homophilic decision-making, such as inequality, segregation and online echo chambers.