Pythagorean triangles of equal areas. (1988) - Baica, Malvina
Abstract : The main intent in this paper is to find triples of Rational Pythagorean Triangles (abbr. RPT) having equal areas. A new method of solvinga 2 + a b + b 2 = c 2 is to seta = y − 1, b = y + 1, y ∈ N − { 0, 1 } and get Pell's equationc 2 − 3 y 2 = 1 . To solvea 2 − a b − b 2 = c 2, we seta = 1 2 ( 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 toa 2 ± a b + b 2 = c 2 . From this fact the following theorems are proved. Theorem 1 Letc 2 = a 2 + a b + b 2, a + b > c > b > a > 0, then the three RPT-s formed by( c, a ), ( c, b ), ( a + b, c ) have the same areaS = a b c ( b + a ) and there are infinitely many such triples of RPT. Theorem 2 Letc 2 = a 2 − a b + b 2, b > c > a > 0, then the three RPT-s formed by( b, c ), ( c, a ), ( c, b − a ) have the same areaS = a b c ( b − a ) and there are infinitely many such triples of RPT.