A cornucopia of pythagorean triangles

As with many works in mathematics, the origin of this article lies in a discussion with a colleague about a classroom question. In Figure 1, two circles are illustrated, centered at 1 C and 2 C . The two circles have only one point of intersection I ; that point I being the point of tangency between the two circles; they two circles being externally tangential. The two circles have three tangents or tangent lines in common. These are the two congruent tangents 2 1 T T and 1 T  2 T  , as well as their third common tangent which is the line perpendicular to This, as it turns out, is a simple matter, and the answer (as we will see below) is

. This then led to a number theoretic exploration by this author, which resulted in this work. If we look at Figure 1, we can identify sixteen right triangles, which are listed in Section 2. Now, consider the case when the two radii 1 R and 2 R are integers. In Section 6, we give the precise conditions the radii 1 R and 2 R must satisfy in order that all the sixteen right triangles ( listed in Section 2) are actually Pythagorean. In Section 3, we offer some immediate geometric observations from Figure 1, and in Section 4 we compute the side lengths of these Pythagorean triangles.

In Section 5, we state three results form number theory ( we offer a proof for Result 3) ; including the well known parametric formulas which describe the entire family of Pythagorean triples ( Result 1). In Section 7, we list the sidelengths of the 16 Pythagorean triangles and in Section 8 we present a numerical example. Finally, in Section 9, we explain why the diagonal lengths 2) We will denote by XY the full straight line that goes through the two points X and Y ; and by YX , the half-line or ray which emanates from Y ( or whose vertex is the point Y ),and which only contains those points on ( of the full line XY ) which lie on the side of (the point Y ) which contains X .

These are (from Figure 1) :

The two congruent right triangles

Looking at Figure 1, we see that the geometry involved is pretty obvious and easy. First consider the various angels. We have,

In this section we compute the side lengths of the 16 right triangles (listed in Section 2), in terms of the radii 1 R and 2 R .

The key calculations are those of the lengths . , , , ,

The rest of the lengths follow easily from these five lengths.

Let F be the foot of the perpendicular from the point 2 C to the line segment

And from the right triangle

From the right triangle IM C

Working similarly, from the right triangle

Next, from the right triangles

By solving the linear system of equations ( 4) and (5) in 2 1 a and 2 2 a (may use Kramer’s Rule).

We further obtain Likewise,

, as it can be easily seen from ( 7b), (7d) , (6c), and (6d). This can also be seen from the fact that the quadrilateral 2 1 MM IM is a rectangle.

To finish the length computations; we must compute the lengths

From the similarity of the right triangles

or alternatively, by using the similarity of the triangles

; and by (1) we obtain, after solving for K T

The parametric formulas listed below in Results 1, describe the entire family of Pythagorean triples. A wealth of historical information on Pythagorean triangles can be found in references

such that m and n are relatively prime, n m  , and n m, have different parities (i.e. one of them is even, the other odd) When

, the Pythagorean triple is called primitive. Also, note that

The following result is well known and it can easily be found in number theory books; for example, see c . We obtain Before we proceed further, note that by inspection we see from Figure 1 and because of the rectangle

; and the rationality of the lengths ,,,,,,  ; are rational numbers. Next, we calculate the above lengths in terms of the integers 2 1 , , r r  and 3 r . The computations are straightforward. One simply uses (10) in conjunction with (6a)-(6d), (7a)-(7d), ( 8) and (9a)-(9d) in order to obtain the following: The following coprimeness conditions follow readily (from ( 10) ) ( may also use

A fundamental result in number theory is Euclid’s Lemma, which says that if an integer a divides the product bc and a is relatively prime to b , then a must be a divisor of the integer c

. Applying Euclid’s Lemma to the formulas in (11d)-(11i) and (11k)-(11n) and in conjunction with the coprimeness conditions (12) the following becomes clear: In order that the lengths

; be integers as well, it is necessary and sufficient that the integer  be divisible by both

Consequently, from (11a)-(11n) and ( 13) we obtain the following length formulas:

Below we list the sixteen Pythagorean triangles obtained when all the lengths These 16 Pythagorean triangles are:

The two congruent Pythagorean triangles . They have hypotenuse length

, and leg lengths

The two congruent Pythagorean triangles

The two congruent Pythagorean triangles

The first primitive Pythagorean triple  

. We take 1  t , and we apply formulas

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