International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 769-780
doi:10.1155/S0161171288000948
Pythagorean triangles of equal areas
Department of Mathematics and Computer Science, The University of Wisconsin, Whitewater 53190, Wisconsin , USA
Received 28 January 1987; Revised 13 April 1987
Copyright © 1988 Malvina Baica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.