Pythagorean triangles of equal areas

[摘要] The main intent in this paper is to find triples of Rational Pythagorean Triangles (abbr. RPT) having equal areas. A new method of solvinga2+ab+b2=c2is to seta=y−1,b=y+1,y∈N−{0,1}and get Pell's equationc2−3y2=1. To solvea2−ab−b2=c2, we seta=12(y+1),b=y−1,y≥2,y∈Nand get a corresponding Pell's equation. The infinite number of solutions in Pell's equation gives rise to an infinity of solutions toa2±ab+b2=c2. From this fact the following theorems are proved.Theorem 1 Letc2=a2+ab+b2,a+b>c>b>a>0, then the three RPT-s formed by(c,a),(c,b),(a+b,c)have the same areaS=abc(b+a)and there are infinitely many such triples of RPT.Theorem 2 Letc2=a2−ab+b2,b>c>a>0, then the three RPT-s formed by(b,c),(c,a),(c,b−a)have the same areaS=abc(b−a)and there are infinitely many such triples of RPT.

[学科分类] 数学（综合）