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The purpose of this paper is to give a new provement to inequality sin Σ sin A/2 ≤ [square root of 4R+r/2r], who are given in [1], to prove that this is the better inequality of type: Σ sin A/2 ≤ [square root of αR+βr/R], when [square root of αR+βr/R] ≤3/2, (1) and to refine this inequality with an equality of type: 'better of the type': Σ sin A/2 ≤ αR+βr, when αR+βr ≤ 3/22 (2) or in an equivalent from: Σ sin A/2 ≤ [square root of 2]R + (3 - 2 [square root of 2])r/R (3) We denote R/r = x, d = [square root of [R.sup.2] - 2Rr, [d.sub.x] = [square root of [x.sup.2] - 2x]. Keywords Geometrical inequalities. 2010 Mathematics Subject Classification: 26D15, 26D15, 51M16., §1. Main results Lemma 1.1. In all triangle ABC holds: [square root of 2]R + (3 - 2 [square root of 2])r/R [less than or equal to] [square root of [...]

In the paper the authors proposed the following inequality: In any acute triangle are true the following inequality: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1) Keywords Geometric inequalities. 2010 Mathematics Subject Classification: 26D15, 51[M.sub.1] 6, 26D99., [section]1. Main results In any triangle ABC we shall denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the radius of circumcircle and r the radius of incircle. In the following [...]