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The Chernoff information was originally introduced for bounding the probability of error of the Bayesian decision rule in binary hypothesis testing. Nowadays, it is often used as a notion of symmetric distance in statistical signal processing or as a way to define a middle distribution in information fusion. Computing the Chernoff information requires to solve an optimization problem that is numerically approximated in practice. We consider the Chernoff distance for distributions belonging to the same exponential family including the Gaussian and multinomial families. By considering the geometry of the underlying statistical manifold, we define exactly the solution of the optimization problem as the unique intersection of a geodesic with a dual hyperplane. Furthermore, we prove analytically that the Chernoff distance amounts to calculate an equivalent but simpler Bregman divergence defined on the distribution parameters. It follows a closed-form formula for the singly-parametric distributions, or an efficient geodesic bisection search for multiparametric distributions. Finally, based on this information-geometric characterization, we propose three novel information-theoretic symmetric distances and middle distributions, from which two of them admit always closed-form expressions.