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In this work, we establish near-linear and strong convergence for a natural first-order iterative algorithm that simulates Von Neumann's Alternating Projections method in zero-sum games. First, we provide a precise analysis of Optimistic Gradient Descent/Ascent (OGDA) -- an optimistic variant of Gradient Descent/Ascent -- for \emph{unconstrained} bilinear games, extending and strengthening prior results along several directions. Our characterization is based on a closed-form solution we derive for the dynamics, while our results also reveal several surprising properties. Indeed, our main algorithmic contribution is founded on a geometric feature of OGDA we discovered; namely, the limit points of the dynamics are the orthogonal projection of the initial state to the space of attractors. Motivated by this property, we show that the equilibria for a natural class of \emph{constrained} bilinear games are the intersection of the unconstrained stationary points with the corresponding probability simplexes. Thus, we employ OGDA to implement an Alternating Projections procedure, converging to an $ε$-approximate Nash equilibrium in $\widetilde{\mathcal{O}}(\log^2(1/ε))$ iterations. Our techniques supplement the recent work in pursuing last-iterate guarantees in min-max optimization. Finally, we illustrate an -- in principle -- trivial reduction from any game to the assumed class of instances, without altering the space of equilibria.