This paper fully characterizes the geometrical structure of the class of distributions of three-dimensional Bernoulli random variables with equal means, $p$. We find all the geometrical generators in closed form as functions of $p$. This result stems from an algebraic representation of the class that encodes the statistical properties of Bernoulli distributions. We study extremal negative dependence within the class and provide an application example by finding the impact of negative dependence to minimal aggregate risk. The application relies on a game theory approach.