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 F be a figure of area A with one side of
length 1, and consider a right triangle with
sides of lengths a, b, and c with the side of
length c the hypotenuse. Draw scaled versions
of the figure F on each of the three sides of the
triangle. Then, as area scales as length squared,
the area of the large figure is a
2
A, and the
areas of the two smaller copies are b
2
A and
c
2
A. By the ordinary version of Pythagoras’s
theorem we have c
2
A = a
2
A + b
2
A.
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 c labeled C, the following relation holds:
c
2 = a
2 + b
2 – 2ab cosC
Thus if an arbitrary triangle were to satisfy Pythagoras’s relation c
2 = a
2 + b
2
, then it must be the case that
cosC equals zero, meaning that C is a 90° angle. This
gives the converse to Pythagoras’s theorem:
If an arbitrary triangle with side lengths a, b,
and c satisfies a
2 + b
2 = c
2
, 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 3
2 +
4
2 = 5
2
. (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 alternative proof of the Pythagorean converse.
Any set of integers a, b, and c, satisfying the relation a
2 + b
2 = c
2 is called a PYTHAGOREAN 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 a
2 + b
2 = c
2 is called a
Pythagorean triple. For example, (3,4,5) and (48,55,73)
are two sets of Pythagorean triples. The converse of
PYTHAGORAS’S THEOREM shows that any Pythagorean
triple (a,b,c) corresponds to the side-lengths of a right triangle, with the side of length c as hypotenuse. Egyptian
architects, for example, used knotted ropes to create 3-45 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 Babylonian era (ca. 1600 B.C.E.), contains a table of right triangles 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 uninteresting and scholars tend to focus on those triples (a,b,c)
for which the numbers a, b, and c share no common factors (other than the number 1). Such Pythagorean triples
are called primitive. For example, (5,12,13) is a primitive Pythagorean triple, but (60,63,87) is not.
The Greek mathematician EUCLID (ca. 300 B.C.E.)
in book 10 of his text THE ELEMENTS completely classified the primitive Pythagorean triples. He showed that
any primitive Pythagorean triple must be of the form:
(p
2 – q
2
, 2pq, p
2 + q
2
)
for some positive integers p and q, one even, one odd,
with p > q and sharing no common factors. For example, the primitive triple (3,4,5) is obtained by setting p
= 2 and q = 1. His proof of this used only basic principles of arithmetic but was somewhat complicated.
Using the theory of COMPLEX NUMBERS, however, the
Swiss mathematician LEONHARD EULER discovered in
the 1700s an elementary derivation of this result:
Suppose a, b, and c are three integers satisfying the equation c
2 = a
2 + b
2 . Write a
2 + b
2 =
(a + ib)(a – ib). Since every complex number
426 Pythagorean triples
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 F be a figure of area A with one side of
length 1, and consider a right triangle with
sides of lengths a, b, and c with the side of
length c the hypotenuse. Draw scaled versions
of the figure F on each of the three sides of the
triangle. Then, as area scales as length squared,
the area of the large figure is a
2
A, and the
areas of the two smaller copies are b
2
A and
c
2
A. By the ordinary version of Pythagoras’s
theorem we have c
2
A = a
2
A + b
2
A.
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 c labeled C, the following relation holds:
c
2 = a
2 + b
2 – 2ab cosC
Thus if an arbitrary triangle were to satisfy Pythagoras’s relation c
2 = a
2 + b
2
, then it must be the case that
cosC equals zero, meaning that C is a 90° angle. This
gives the converse to Pythagoras’s theorem:
If an arbitrary triangle with side lengths a, b,
and c satisfies a
2 + b
2 = c
2
, 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 3
2 +
4
2 = 5
2
. (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 alternative proof of the Pythagorean converse.
Any set of integers a, b, and c, satisfying the relation a
2 + b
2 = c
2 is called a PYTHAGOREAN 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 a
2 + b
2 = c
2 is called a
Pythagorean triple. For example, (3,4,5) and (48,55,73)
are two sets of Pythagorean triples. The converse of
PYTHAGORAS’S THEOREM shows that any Pythagorean
triple (a,b,c) corresponds to the side-lengths of a right triangle, with the side of length c as hypotenuse. Egyptian
architects, for example, used knotted ropes to create 3-45 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 Babylonian era (ca. 1600 B.C.E.), contains a table of right triangles 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 uninteresting and scholars tend to focus on those triples (a,b,c)
for which the numbers a, b, and c share no common factors (other than the number 1). Such Pythagorean triples
are called primitive. For example, (5,12,13) is a primitive Pythagorean triple, but (60,63,87) is not.
The Greek mathematician EUCLID (ca. 300 B.C.E.)
in book 10 of his text THE ELEMENTS completely classified the primitive Pythagorean triples. He showed that
any primitive Pythagorean triple must be of the form:
(p
2 – q
2
, 2pq, p
2 + q
2
)
for some positive integers p and q, one even, one odd,
with p > q and sharing no common factors. For example, the primitive triple (3,4,5) is obtained by setting p
= 2 and q = 1. His proof of this used only basic principles of arithmetic but was somewhat complicated.
Using the theory of COMPLEX NUMBERS, however, the
Swiss mathematician LEONHARD EULER discovered in
the 1700s an elementary derivation of this result:
Suppose a, b, and c are three integers satisfying the equation c
2 = a
2 + b
2 . Write a
2 + b
2 =
(a + ib)(a – ib). Since every complex number
426 Pythagorean triples