In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers which are the leglengths of a Pythagorean triangle. Using the idea of Pythagorean rationals, we generate two families of rational triangles. We define a rational triangle to be a triangle with rational sidelengths and area.
The purpose of this paper is to present a simple method for generating rational triangles. These are triangles with rational side lengths and rational area. We do this in Section 5, wherein we generate a certain family of such triangles. In Section 2, we present some well known triangle formulas involving a triangle's side lengths, heights, area, the Law of Sines, and the Law of Cosines. Also two formulas for R and r , the radius of a triangle's circumscribed circle and the radius of its inscribed circle respectively. We offer a short derivation of these two formulas in Section 7. In Section 3, we state the well known parametric formulas for Pythagorean triples, as well as a table featuring a few Pythagorean triples. In Section 4 we state two definitions; that of a rational triangle and the definition of a Pythagorean rational. And in Section 6, we offer two closing remarks. (1) The next two formulas are not as well known to a wide mathematical audience as the first four.
We present a short derivation for these two formulas in Section 7.
A Pythagorean triple (a, b, c) of positive integers a, b, c; is one which satisfies 1 2 1 3 4 5 1 3 2 5 12 13 1 4 1 15 8 17 1 4 3 7 24 25 1 5 2 21 20 29 1 5 4 9 40 41 Table 1 (7
By inspection of formuals ( 1) -( 6) the following is evident.
If , , a b g are rational, then the three cosine values cos , cos , cos A B G , are always rational, regardless of whether the triangle ABG is rational.
If ABG is a rational triangle, then the three heights, the three sine values, and the radii R and r are also rational.
If ABG is not a rational triangle, then E is an irrational number, and consequently the three sine values, the three heights, and the two radii are all irrational numbers.
Remark 4: Note that a Pythagorean rational as defined in Definition 1, is always a proper rational; that is, it cannot be an integer. This can be easily seen from the conditions and formulas in (7). On the other hand, the notion of a Pythagorean number, is often found in the literature of number theory.
A Pythagorean number is a number which is equal to the area of a Pythagorean triangle. Thus, by (7), it is always an integer equal to 2 2 2 . .( ) mn m n d a Pythagorean number is never a proper rational. (1, ) A m on the first and second rays respectively; T(l, 0) on the positive o>axis. A OA ; 0 < 2 w < 90 Also for side lengths:
Straightforward calculations yield the following results.
Heights
Cosine and Sine Values and R r
(To obtain these we use ( 5) and ( 6), combined with ( 8) and ( 9))
A glance at formulas (7) with
Thus, if we take n = 1 (and m even), it is obvious that the Pythagorean rationals a b can be arbitrarily large; while b a arbitrarily small. Therefore in Figure 1, either 1 m or m 2 can be arbitrarily large or small; and so the degree measure j of the angle 1 2 A OA can be arbitrarily close to 180 or to 0.
The derivation of ( 5) is very short indeed. From (2) and (3) we obtain
The derivation of ( 6) is also very short, if we just take a look at Figure 2
. Thus, by
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