On the fundamental triangle inequality and Gerretsen’s double inequality

We give two new proofs of the fundamental triangle inequality. By using the fundamental triangle inequality and the related Blundon’s theorem, we prove several new general results. We also obtain two new generalizations of Gerretsen’s double inequality, from which some new and old refinements of Gerretsen’s double inequality are obtained as well. In addition, we propose a conjecture related to the fundamental triangle inequality.