PYTHAGOREAN TRIANGLES OF EQUAL AREAS

MALVINA BAICA

《International Journal of Mathematics and Mathematical Sciences》 1988卷 (1988/12) Pp. 769-780

https://doi.org/10.1155/S0161171288000948

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 solving a^2+ab+b^2=c^2 is to set a=y-1, b=y+1, y∈N - {0,1} and get Pell's equation c^2-3y^2=1. To solve a^2−ab−b^2=c^2, we set a=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 to a^2±ab+b^2=c^2. From this fact the following theorems are proved. Theorem 1 Let c^2=a^2+ab+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 area S=abc(b + a) and there are infinitely many such triples of RPT. Theorem 2 Let c^2=a^2−ab+b^2, 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.

關鍵字

Rational Pythagorean Triangles (abbr. RPT) ； Perimeter of the RPT ； Pell's (Euler's) equation ； Fibonacci sequence