Since Nesterov's seminal 1983 work, many accelerated first-order optimization\nmethods have been proposed, but their analyses lacks a common unifying\nstructure. In this work, we identify a geometric structure satisfied by a wide\nrange of first-order accelerated methods. Using this geometric insight, we\npresent several novel generalizations of accelerated methods. Most interesting\namong them is a method that reduces the squared gradient norm with\n$\\mathcal{O}(1/K^4)$ rate in the prox-grad setup, faster than the\n$\\mathcal{O}(1/K^3)$ rates of Nesterov's FGM or Kim and Fessler's FPGM-m.\n