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众所周知,给出三角形的三边a、b、c的长,三角形的面积S就可用秦九韶公式或海伦公式求出。如果三角形的三边a,b,c都是自然数,由于以上两个面积公式都带有根号,所以求出的面积未必是自然数。特别当三边为连续自然数时,面积也未必是自然数。本文要解决的问题是:三角形的三边是怎样的连续自然数时,面积也是自然数,并求出一切这样的三角形。
As we all know, given the length of the three sides a, b, and c of the triangle, the area S of the triangle can be calculated using the formula of Qin Jiuhao or Helen’s formula. If the three sides a, b, and c of the triangle are all natural numbers, since the above two area formulas all have root numbers, the calculated area is not necessarily a natural number. Especially when the three sides are continuous natural numbers, the area is not necessarily a natural number. The problem to be solved in this paper is: when the three sides of a triangle are continuous natural numbers, the area is also a natural number, and all such triangles are found.