Abstract <p>The article introduces a characteristic of a triangle, reflecting the measure of its nondegeneracy. The importance of studying this quantity is associated with the construction of high-quality computational grids. It is shown that if a Sobolev class mapping distorts this characteristic multiple times, then this mapping is a mapping with bounded distortion. In addition, it is proved that if the above condition and additionally the condition of bounded distortion of the area of the triangle are satisfied, then the mapping is bi-Lipschitz. The article establishes estimates for all constants characterizing the mappings under study.</p>