the triangles are arranged. In particular, the left and
right arrangements show the theorem to be true.
The theorem is named after the Greek philosopher,
mathematician, and mystic P
YTHAGORAS
(ca. 500
B
.
C
.
E
.). Babylonian tablets dating before 1600
B
.
C
.
E
.
listed tables of numbers a, b and csatisfying the rela-
tionship a2+ b2= c2, suggesting that scholars of the
time were aware of the result. (It is believed that the
Babylonians used 3-4-5 triangles to measure right
angles when planning constructions.) The theorem is
stated explicitly in the ancient Hindu text Sulbasutram
(ca. 1100
B
.
C
.
E
.) on temple buildings, and the diagram
of four triangles arranged in a square appears in the
ancient Chinese text Chou Pei Suan Ching of about
500
B
.
C
.
E
Although scholars from other cultures may
have been aware of the result, Pythagoras is credited as
the first to give full and proper explanation as to why
the result is true.
Today many different proofs of Pythagoras’s theo-
rem are known. The ancient Greeks (thinking solely in
terms of geometric constructs) showed how to explic-
itly divide the largest square into four pieces that could
be rearranged to form the two smaller squares. The
method of proof presented above is often called the
Chinese proof. United States President James Garfield
published his own proof of Pythagoras’s theorem in
1876 as part of his test to become a high-ranking
Mason. Early in the 20th century Professor Elisha Scott
Loomis collated and published 367 different demon-
strations of the result in his book The Pythagorean
Proposition. Tiling a floor with squares of two differ-
ent sizes provides a surprising visual proof of the theo-
rem. (See
SQUARE
.) New proofs of this famous result
are still being discovered today.
Consequences of the Theorem
If a right triangle has side-lengths a, b, and c, with the
side of length copposite the right angle, then Pythago-
ras’s theorem asserts c2= a2+ b2, from which it follows
that cis larger than both aand b. Thus:
In a right triangle, the side opposite the right
angle is indeed the hypotenuse of the triangle.
It is longer than either of the remaining two
sides.
Although this observation seems trivial, it has some
important consequences:
1. The shortest distance dof a point Pfrom a line Lis
given by the length of the perpendicular from Pto L.
In the diagram above, any other line segment
connecting Pto Lis longer than d.
This observation allows one to prove all of the
standard
CIRCLE THEOREMS
.
2. In any triangle, the sum of the lengths of any two
sides is larger than the length of the third.
This result, called the
TRIANGULAR INEQUALITY
,
follows by drawing a perpendicular line from the
apex of the triangle. In the diagram above right, we
have that a+ bis greater than c1+ c2= c.
3. The shortest distance between two points in a plane
is given by a straight line.
The triangular inequality shows that the line
directly connecting two vertices of a triangle is
shorter than the sum of the two remaining side-
lengths. Thus, if a path connecting points Aand Bis
composed of straight-line segments, each pair of seg-
ments can be replaced with a shorter single straight-
line segment. Repeated application of this procedure
eventually replaces the path with the straight-line
path connecting Ato B. If, on the other hand, a
path connecting Ato Bis curved, then one can
approximate the curved path by one composed
solely of small straight-line segments. As we have
just seen, the direct straight-line path connecting A
to Bis shorter than any such approximation. One
can argue that if the approximation is made with
some degree of precision, the straight-line path con-
necting Ato Bis shorter than the curved path too.
Generalized Pythagorean Theorem
The shapes constructed on the sides of a right triangle
need not be squares for Pythagoras’s theorem to hold
true. For example, if one were to construct equilateral
triangles on each of the three sides of a right triangle,
then the area of the large triangle would equal the sum
Pythagoras’s theorem 425
Consequences of Pythagoras’s theorem