Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1982. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 16)

II. TANULMÁNYOK A TERMÉSZETTUDOMÁNYOK KÖRÉBÖL - H. Molnár Sándor: Háromszögek szögeinek lineáris függetlenségéről

ON THE LINEAR INDEPENDENCE OF ANGLES OF TRIANGLES by Sándor Molnár (Summary) Let a

i

5 bi, c

i (i = 1, 2, . . n) be sides of rectangular triangles so that — a

l

9 bi, c

i are integers and Cj, a

i

? bj are coprime — c'j s are distinct prime integers — 0i is one of the acute angles of a triangle with sides a

i

; bi, q. It is shown in [2] that 0

i (i = 1, 2,. . ., n) are linearly independent over the rational field. We studied this problem in [3] when Ci' s are not necessarily prime integers. In this paper — in case n = 2 — we give a necessary and sufficient condition for the sides so that Q\ s are linearly dependent over the rationals. If the q' s are not necessarily integers then we prove a sufficient condition for the sides so that @i (i = 1, 2) are linearly independent over the rationals. Among others we prove the following result. Let ai, bi, ci and a2, bo, c? be sides of two arbitrary distinct triangles, let ai, b

h q (i = 1,2) be integers and where (si, ti) = 1 (i = 1, 2). If there is an odd prime integer p, which divides exactly one of t; and to, then Si and (9o are linearly independent over the rationals. 0\ = arc cos > 2 . 2 2 bi +Cj — a j 2bi C

i U 574