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Pythagorean Theorem
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For a right triangle with legs and and hypotenuse ,
|
(1)
|
Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a plane triangle
to a trirectangular tetrahedron, in
which case it is known as de Gua's theorem. The
various proofs of the Pythagorean theorem all seem to require application of some
version or consequence of the parallel postulate:
proofs by dissection rely on the complementarity of the acute angles of the right
triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs
by similarity require the existence of non-congruent similar triangles, and so on
(S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean
theorem.
After receiving his brains from the wizard in the 1939 film The Wizard of Oz, the Scarecrow recites the following mangled (and incorrect) form of the Pythagorean
theorem, "The sum of the square roots of any two sides of an isosceles
triangle is equal to the square root of the remaining side." In the fifth
season of the television program The Simpsons, Homer J. Simpson repeats
the Scarecrow's line (Pickover 2002, p. 341). In the Season 2 episode "Obsession"
(2006) of the television crime drama NUMB3RS,
Charlie's equations while discussing a basketball hoop include the formula for the
Pythagorean theorem.
A clever proof by dissection which reassembles two small squares into one larger one was given by the Arabian mathematician Thabit ibn
Kurrah (Ogilvy 1994, Frederickson 1997).
Another proof by dissection is due to Perigal (left figure; Pergial 1873; Dudeney 1958; Madachy 1979; Steinhaus 1999, pp. 4-5; Ball
and Coxeter 1987). A related proof is accomplished using the above figure at right,
in which the area of the large square
is four times the area of one of the triangles
plus the area of the interior square.
From the figure, ,
so
The Indian mathematician Bhaskara constructed a proof using the above figure, and another beautiful dissection proof is shown below (Gardner 1984, p. 154).
|
(7)
|
|
(8)
|
|
(9)
|
Several beautiful and intuitive proofs by shearing exist (Gardner 1984, pp. 155-156;
Project Mathematics!).
Perhaps the most famous proof of all times is Euclid's geometric proof (Tropfke 1921ab; Tietze 1965, p. 19), although it is neither the simplest nor the most obvious. Euclid's proof used the figure below, which is sometimes known variously as the bride's chair, peacock tail, or windmill. The philosopher Schopenhauer has described this proof as a "brilliant piece of perversity" (Schopenhauer 1977; Gardner 1984, p. 153).
Let
be a right triangle, , , and be squares, and . The triangles and are equivalent except for rotation, so
|
(10)
|
Shearing these triangles gives two more equivalent triangles
|
(11)
|
Therefore,
|
(12)
|
Similarly,
|
(13)
|
so
|
(14)
|
Heron proved that ,
,
and intersect in a point (Dunham 1990, pp. 48-53).
Heron's formula for the area of the triangle, contains the Pythagorean theorem implicitly.
Using the form
|
(15)
|
and equating to the area
|
(16)
|
gives
|
(17)
|
Rearranging and simplifying gives
|
(18)
|
the Pythagorean theorem, where is the area of a triangle
with sides ,
,
and
(Dunham 1990, pp. 128-129).
A novel proof using a trapezoid was discovered by James Garfield (1876), later president of the United States, while serving in the House
of Representatives (Gardner 1984, pp. 155 and 161; Pappas 1989, pp. 200-201;
Bogomolny).
Rearranging,
|
(22)
|
|
(23)
|
|
(24)
|
An algebraic proof (which would not have been accepted by the Greeks) uses the Euler formula. Let the sides of a trianglebe,
,
and ,
and the perpendicular legs of right
triangle be aligned along the real and imaginary axes. Then
|
(25)
|
Taking the complex conjugate gives
|
(26)
|
Multiplying (25) by (26) gives
|
(27)
|
(Machover 1996).
Another algebraic proof proceeds by similarity. It is a property of right triangles, such as the one shown in the above left figure, that the right
triangle with sides , , and (small triangle in the left figure; reproduced in the right
figure) is similar to the right triangle with sides
,
,
and
(large triangle in the left figure; reproduced in the middle figure). Letting in the above left figure then gives
so
and
|
(32)
|
(Gardner 1984, p. 155 and 157). Because this proof depends on proportions of potentially irrational numbers and cannot be
translated directly into a geometric construction,
it was not considered valid by Euclid.
See also
Bride's Chair, de Gua's Theorem, Law of Cosines, Peacock
Tail, Pythagoras's Theorem, Pythagorean
Triple, Right Triangle, Windmill Explore this
topic in the MathWorld classroom
Explore with Wolfram|Alpha
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References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 87-88, 1987.Bogomolny,
A. "Pythagorean Theorem." http://www.cut-the-knot.org/pythagoras/index.shtml.Brodie,
S. E. "The Pythagorean Theorem Is Equivalent to the Parallel Postulate."
http://cut-the-knot.org/triangle/pythpar/PTimpliesPP.html.Dixon,
R. "The Theorem of Pythagoras." §4.1 in Mathographics.
New York: Dover, pp. 92-95, 1991.Dudeney, H. E. Amusements
in Mathematics. New York: Dover, p. 32, 1958.Dunham, W.
"Euclid's Proof of the Pythagorean Theorem." Ch. 2 in Journey
through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990.Frederickson,
G. Dissections:
Plane and Fancy. New York: Cambridge University Press, pp. 28-29, 1997.Friedrichs,
K. O. From
Pythagoras to Einstein. Washington, DC: Math. Assoc. Amer., 1965.Gardner,
M. "The Pythagorean Theorem." Ch. 16 in The
Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, pp. 152-162, 1984.Garfield, J. A. "Pons
Asinorum." New England J. Educ. 3, 161, 1876.Kern,
W. F. and Bland, J. R. Solid
Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948.Loomis,
E. S. The
Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography
of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston,
VA: National Council of Teachers of Mathematics, 1968.Machover, M. "Euler's
Theorem Implies the Pythagorean Proposition." Amer. Math. Monthly 103,
351, 1996.Madachy, J. S. Madachy's
Mathematical Recreations. New York: Dover, p. 17, 1979.Ogilvy,
C. S. Excursions
in Mathematics. New York: Dover, p. 52, 1994.Pappas, T.
"The Pythagorean Theorem," "A Twist to the Pythagorean Theorem,"
and "The Pythagorean Theorem and President Garfield." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 4, 30,
and 200-201, 1989.Parthasarathy, K. R. "An-Dimensional Pythagoras Theorem." Math. Scientist 3,
137-140, 1978.Perigal, H. "On Geometric Dissections and Transformations."
Messenger Math. 2, 103-106, 1873.Pickover, C. A.
"The Scarecrow Formula." Ch. 103 in The
Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge
University Press, pp. 217-218 and 341, 2002.Project Mathematics.
"The Theorem of Pythagoras." Videotape. http://www.projectmathematics.com/pythag.htm.Schopenhauer,
A. The
World as Will and Idea, 3 vols. New York: AMS Press, 1977.Shanks,
D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 123-127,
1993.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, 1999.Talbot, R. F.
"Generalizations of Pythagoras' Theorem in Dimensions." Math. Scientist 12, 117-121,
1987.Tietze, H. Famous
Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity
to Modern Times. New York: Graylock Press, p. 19, 1965.Tropfke,
J. Geschichte der Elementar-Mathematik, Band 1. Berlin: p. 97, 1921a.Tropfke,
J. Geschichte der Elementar-Mathematik, Band 4. Berlin: pp. 135-136,
1921b.Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 202-207, 1991.Yancey, B. F. and Calderhead, J. A.
"New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 3,
65-67, 110-113, 169-171, and 299-300, 1896.Yancey, B. F. and Calderhead,
J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math.
Monthly 4, 11-12, 79-81, 168-170, 250-251, and 267-269, 1897.Yancey,
B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean
Theorem." Amer. Math. Monthly 5, 73-74, 1898.Yancey,
B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean
Theorem." Amer. Math. Monthly 6, 33-34 and 69-71, 1899.
Cite this as:
Weisstein, Eric W. "Pythagorean Theorem."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PythagoreanTheorem.html
Subject classifications
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Geometry
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Plane Geometry
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Triangles
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Triangle Properties
●
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Mathematics in the Arts
●
Mathematics in Films
●
The Wizard of Oz (1939)
●
Recreational Mathematics
●
Mathematics in the Arts
●
Mathematics in Television
●
NUMB3RS
●
Recreational Mathematics
●
Mathematics in the Arts
●
Mathematics in Television
●
The Simpsons
More...Less...
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