Refined geometric inequalities between two or more triangles obtained by dedublation

摘要

We study a class of inequalities between two or more triangles which extend the known metric relations between the elements of a single triangle. The common idea is that any quadratic type inequality between the elements of one triangle can have a "dedublated form" when written between the elements of two (or more) triangles with the optimal inequality being possible only when the triangles are similar. For example, we extend the well known quadratic form inequalities of Gerretsen [2, page 8] and give the new, dedublated form inequalities for the relations, which, in the case of a single triangle, correspond to the distances between the important points of the triangle such as circumcentre, incentre, orthocentre and the centre of mass.