一个常见三角不等式的推广

• 作 者: 宋庆
• 机 构: 南昌大学附中 江西330047
• 出 处: 数学通讯
• 发表日期: 2006年23期
• 关键词: 三角不等式 当且仅当 正整数

在△ABC,有不等式cosAcosBcosC≤81(1)等号成立当且仅当△ABC为正三角形.将其推广,笔者获得如下结论.定理在△ABC中,对λ≥0有不等式cosAcosB(cosC+λ)≤(1+8λ)2(2)等号成立当且仅当A=B=21arccosλ2-1.证当cosAcosB≤0时,cosC0,从而cosAcosB(cosC+λ)≤0(1+8λ)2;当cosAcosB

In △ABC, there is an inequality cosAcosBcosC≤81(1) The equal sign holds if and only if △ABC is a regular triangle. To generalize it, the author obtains the following conclusion. Theorem In △ABC, there is an inequality for λ≥0 cosAcosB(cosC+ λ) ≤ (1+8λ) 2 (2) equal sign holds if and only if A = B = 21arccosλ2-1. When cosAcosB ≤ 0, cosC0, and thus cosAcosB (cosC + λ) ≤ 0 (1+8λ) 2; when cosAcosB