Abstract

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