定理若a,b,c∈R,则a~2+b~2+c~2≥ab+bc+ca,当且仅当a=b=c时等号成立.证明由aa~2+b~2≥2ab,b~2+c~2≥2bc,c~2+a~2≥2ca相加后除以2即得定理中的不等式.以上三个不等式分别在a=b,b=c,c=a时等号成立,故当且仅当a=b=c时取等号.定理中的不等式虽然平凡,但却有着广泛的应用价值,它在处理“三项和≥另三项和”型问题时,常有上佳表现.下面我们结合例题予以说明,供同学们学习揣摩. Theorem If a, b, c∈R, then a ~ 2 + b ~ 2 + c ~ 2≥ab + bc + ca, if and only if a = b = ~ 2≥2ab, b ~ 2 + c ~ 2≥2bc, c ~ 2 + a ~ 2≥2ca and then divided by 2 to get the inequality in the theorem.The above three inequalities are respectively shown as a = b, b = c , the equal sign holds when c = a, so if and only if a = b = c, take the equal sign. Although the inequality in the theorem is trivial, it has wide application value. Items and "type of problem, often have good performance below we combine examples to explain for the students to learn to figure out.