本文意在从三角形三边的不等式关系、三角形的余弦定理、三角形边长与正实数之间的代换关系出发,探究一道常见的关于三角形三边长不等式的多种证法。1问题呈现设a、b、c为△ABC的三条边,求证:a~2+b~2+c~2<2(ab+bc+ca)。①2证法探究2.1利用三角形三边关系证明我们知道,三角形任两边之和大于第三边,任两边之差小于第三边,由此容易获得上述不等式的直接 This article intends to explore a common set of multiple-syndromes about the triangular long-side inequality from the perspective of the inequality of the three sides of the triangle, the cosine theorem of the triangle, the length of the triangle and the positive real number. Suppose that a, b and c are the three sides of △ ABC. Suppose: a ~ 2 + b ~ 2 + c ~ 2