of the areas of the two small triangles. The same is true
if one were to construct pentagons, semicircles, or any
figure on each side of the triangle, as long as the three
figures are similar. The reason is straightforward:
Let Fbe a figure of area Awith one side of
length 1, and consider a right triangle with
sides of lengths a, b, and cwith the side of
length cthe hypotenuse. Draw scaled versions
of the figure Fon each of the three sides of the
triangle. Then, as area scales as length squared,
the area of the large figure is a2A, and the
areas of the two smaller copies are b2Aand
c2A. By the ordinary version of Pythagoras’s
theorem we have c2A= a2A+ b2A.
The Converse of the Theorem
The
LAW OF COSINES
states that, for any triangle, not
necessarily a right triangle, with side-lengths a, b, and
c, with the angle opposite side clabeled C, the follow-
ing relation holds:
c2= a2+ b2– 2ab cosC
Thus if an arbitrary triangle were to satisfy Pythago-
ras’s relation c2= a2+ b2, then it must be the case that
cosCequals zero, meaning that Cis a 90°angle. This
gives the converse to Pythagoras’s theorem:
If an arbitrary triangle with side lengths a, b,
and csatisfies a2+ b2= c2, then that triangle is
a right triangle (with the right angle opposite
the side of length c).
Thus, for example, a triangle with side lengths 3, 4,
and 5, is indeed a right triangle precisely because 32+
42= 52. (Many elementary texts in mathematics state
without explanation that a 3-4-5 triangle is right.) A
study of the properties of acute, obtuse, and right
angles in a
TRIANGLE
provides a very elementary alter-
native proof of the Pythagorean converse.
Any set of integers a, b, and c, satisfying the rela-
tion a2+ b2= c2is called a P
YTHAGOREAN TRIPLE
. The
triples 5, 12, 13 and 20, 21, 29 are Pythagorean
triples and do indeed form the side-lengths of two
right triangles.
See also
DISTANCE FORMULA
;
SCALE
;
SIMILAR
FIGURES
.
Pythagorean triples A set of positive integers (a,b,c)
satisfying the equation a2+ b2= c2is called a
Pythagorean triple. For example, (3,4,5) and (48,55,73)
are two sets of Pythagorean triples. The converse of
P
YTHAGORAS
’
S THEOREM
shows that any Pythagorean
triple (a,b,c) corresponds to the side-lengths of a right tri-
angle, with the side of length cas hypotenuse. Egyptian
architects, for example, used knotted ropes to create 3-4-
5 triangles and thereby accurately measure 90°angles.
The problem of finding Pythagorean triples is an
ancient one. The oldest record known to exist on the
topic of number theory, a clay tablet from the Babylo-
nian era (ca. 1600
B
.
C
.
E
.), contains a table of right trian-
gles with integer sides. That the triple (4961,6480,8161)
is listed suggests that the Babylonians had a general
method for generating Pythagorean triples and did
not rely on trial and error alone to find them. (It also
suggests that scholars of the time were also interested
in pursuing mathematics simply for the enjoyment of
the subject.)
Multiples of any given Pythagorean triple give new
triples. For example, from the triple (3,4,5) we obtain
new triples (6,8,10), (9,12,15), (12,16,20) and the like.
In some sense, these new Pythagorean triples are uninter-
esting and scholars tend to focus on those triples (a,b,c)
for which the numbers a, b, and cshare no common fac-
tors (other than the number 1). Such Pythagorean triples
are called primitive. For example, (5,12,13) is a primi-
tive Pythagorean triple, but (60,63,87) is not.
The Greek mathematician E
UCLID
(ca. 300
B
.
C
.
E
.)
in book 10 of his text T
HE
E
LEMENTS
completely classi-
fied the primitive Pythagorean triples. He showed that
any primitive Pythagorean triple must be of the form:
(p2– q2, 2pq, p2+ q2)
for some positive integers pand q, one even, one odd,
with p> qand sharing no common factors. For exam-
ple, the primitive triple (3,4,5) is obtained by setting p
= 2 and q= 1. His proof of this used only basic princi-
ples of arithmetic but was somewhat complicated.
Using the theory of
COMPLEX NUMBERS
, however, the
Swiss mathematician L
EONHARD
E
ULER
discovered in
the 1700s an elementary derivation of this result:
Suppose a, b, and care three integers satisfy-
ing the equation c2= a2+ b2. Write a2+ b2=
(a+ib)(a– ib). Since every complex number
426 Pythagorean triples