Theorem on a Matrix of Right-Angled Triangles

Martin W. Bredenkamp
2013 Applied and Computational Mathematics  
The following theorem is proved: All primitive right-angled triangles (primitive Pythagorean triples) may be defined by a pair of positive integer indices (i,j), where i is an uneven number and j is an even number and have no common factor. The sides of every positive integer right angled triangle are then defined by the indices as follows: For hypotenuse h, uneven leg u and even leg e, h = i 2 + ij + j 2 /2, e = ij + j 2 /2, u = i 2 + ij. This defines an infinite by infinite matrix of right angled triangles with positive integer sides.
doi:10.11648/j.acm.20130202.14 fatcat:fp2fg6oiwffn7dq5qdfci4j2y4