We develop a mixed-integer nonlinear programming (MINLP) approach for the classical Heilbronn triangle problem, demonstrating the capability of modern global optimization solvers to tackle challenging combinatorial geometry problems. A symmetry-breaking strategy based on boundary structure yields a substantially stronger model: for n=9, we compute an epsilon-globally optimal point in 15 minutes on a standard desktop computer, improving upon the previously reported effort of approximately one day. By combining numerical certification with exact symbolic computation, we recover exact coordinates matching all best-known configurations for n<=9, including the n=9 configuration of Comellas and Yebra (2002). An analysis of these configurations reveals the clustering of noncritical triangle areas around a small number of distinct values, suggesting rich underlying algebraic structure. All code and data are publicly available.