Behaviors of welded hollow spherical joints strengthened by unidirectional annular ribs

Refined Euler’s inequalities in plane geometries and spaces

“…Note that the right inequality (2.1) is just the AM-GM inequality (i.e., the arithmetic-geometric means inequality) for numbers a 3 , b 3 , c 3 . We presented three proofs of (2.1), two in [6] and another one in [5], modifying and simplifying proof given in [1]. The algebraic proof from [6] basically reduces to the inequality (2.4) in [6]).…”

“…Proofs in [1] and [5] have more geometric flavor by considering tangent segments from vertices to the incircle. Gerretsen's inequalities can also be proved geometrically by computing distances between the incenter I, centroid G and orthocenter H. Namely, 9IG 2 = s 2 − 16Rr + 5r 2 ≥ 0 and IH 2 = 4R 2 + 4Rr − s 2 ≥ 0.…”

“…In [4] we established non-Euclidean versions of Euler's inequality: tan(R) ≥ 2 tan(r) in spherical and tanh(R) ≥ 2 tanh(r) in hyperbolic geometry (when the triangle has the circumcircle). In [1] the authors reproved our Theorem 1 and tried to adapt it to hyperbolic (and spherical) triangles, but it did not work directly. Instead, we found in [6] the right analogue of our improvement of Euler's inequality in non-Euclidean geometries of the previous results.…”