Unlike other observational signatures in cosmology, the angular-diameter distance d A (z) uniquely reaches a maximum (at z max ) and then shrinks to zero towards the big bang. The location of this turning point depends sensitively on the model, but has been difficult to measure. In this paper, we estimate and use z max inferred from quasar cores: (1) by employing a sample of 140 objects yielding a much reduced dispersion due to pre-constrained limits on their spectral index and luminosity, (2) by reconstructing d A (z) using Gaussian processes, and (3) comparing the predictions of seven different cosmologies and showing that the measured value of z max can effectively discriminate between them. We find that z max = 1.70 ± 0.20an important new probe of the Universe's geometry. The most strongly favoured model is R h = ct, followed by Planck ΛCDM. Several others, including Milne, Einstein-de Sitter and Static tired light are strongly rejected. According to these results, the R h = ct universe, which predicts z max = 1.718, has a ∼ 92.8% probability of being the correct cosmology. For consistency, we also carry out model selection based on d A (z) itself. This test confirms that R h = ct and Planck ΛCDM are among the few models that account for angular-size data better than those that are disfavoured by z max . The d A (z) comparison, however, is less discerning than that with z max , due to the additional free parameter, H 0 . We find that H 0 = 63.4±1.2 km s −1 Mpc −1 for R h = ct, and 69.9 ± 1.5 km s −1 Mpc −1 for ΛCDM. Both are consistent with previously measured values in each model, though they differ from each other by over 4σ. In contrast, model selection based on z max is independent of H 0 .