A Concise Proof of the Triangle Inequality for the Tanimoto Distance and its Equality Conditions

The Tanimoto distance is a widely used distance metric in cheminformatics that satisfies the triangle inequality. Several methods have been proposed to prove this inequality; however, we note that there has been no discussion regarding the conditions under which equality holds, which can be crucial in specific contexts. In this paper, we provide a concise and intuitive proof of the triangle inequality for the Tanimoto distance and, based on this proof, derive the corresponding equality conditions.