89.01 Perfect, and almost perfect, Pythagorean triangles

89.28 When the perimeter divides the area of a triangle

“…We now prove that A(7i) E N. Note that, by Lemma 1, P(T) is even, and so A (T) is even, by equation (2). But, if S (T) is odd, then A (T) must be odd from (1), so S(T) is even.…”

“…Among other things, it was noted in [1] that the perimeter of a Pythagorean triangle, with sides a, b and hypotenuse c, divides the area if, and only if, 4 | (c -a) or 4 | (c -b). The number of Pythagorean triples whose area divided by its perimeter is equal to each n e N was calculated in [2]. In this note we investigate this question for arbitrary integer-sided triangles.…”