Yurii N. Maltsev and Anna S. Kuzmina , An improvement of Birsan's inequalities for the sides of a triangle,
Forum Geometricorum, 16 (2016) 81--84.

Abstract. In terms of the distance d between the circumcenter and the incenter of a triangle, B\^{\i}rsan's inequalities can be rewritten as 4((R-d)^2-r^2) <= a^2 <= 4((R+d)^2-r^2), where R and r are respectively the circumradius and inradius, and a is the length of a side of the triangle. We improve these inequalities by finding the best lower and upper bounds of a^2.

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