裴多不等式的解析证明

• 作 者: 陆兆兴 王水汀
• 机 构: 兰州商学院,兰州商学院,
• 出 处: 数学教学研究
• 发表日期: 1989年03期
• 关键词: 证法 当且仅当 距离公式 几何平均值 绝对值符号 算术平均值 等比定理 充分必要条件 成比例 第二象限

裴多(Pedoe)不等式是指:设△ABC,与△A′B′C′的边长分别为α,β,γ与α′β′,γ′,面积分别为S与S′,那么α′~2(β~2+γ~2-α~2)+β′~2(γ~2+α~2-β~2)+γ′~2(α~2+β~2-γ~2)≥16SS′①其中等号当且仅当△ABC～△A′B′C′时成立。这个不等式自1979年传入我国后,以其外形的有趣对称,证法的多种多样引起了广泛的兴趣和讨论,出现了一些新的证法。本文用解析几何的方法给出它的证明。如图放置△ABC与△A′B′C′,则A与

The Pedoe inequality means that the lengths of △ABC and △A′B′C′ are α, β, γ and α′β′, γ′, and the areas are S and S′ respectively. ′~2(β~2+γ~2-α~2)+β′~2(γ~2+α~2-β~2)+γ′~2(α~2+β~2-γ~ 2) ≥ 16SS’1 where the equal sign holds if and only if △ABC to △A’B’C’. Since this inequality was introduced into China in 1979, it has attracted a lot of interest and discussion with its interesting symmetry and various forms of evidence. There have been some new proofs. This paper presents its proof using analytic geometry. As shown in Figure △ABC and △A’B’C’, A and