Proofs
This theorem may have more known proofs than any other (the law of
quadratic reciprocity being also a contender for that distinction); the book
Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs.
Some arguments based on
trigonometric identities (such as
Taylor series for
sine and
cosine) have been proposed as proofs for the theorem. However, since all the fundamental trigonometric identities are proved using the Pythagorean theorem, there cannot be any trigonometric proof. (See also
begging the question.)
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Proof using similar triangles
http://en.wikipedia.org/wiki/Image:Proof-Pythagorean-Theorem.svghttp://en.wikipedia.org/wiki/Image:Proof-Pythagorean-Theorem.svg
Proof using similar triangles
Like many of the proofs of the Pythagorean theorem, this one is based on the
proportionality of the sides of two
similar triangles.
Let
ABC represent a right triangle, with the right angle located at
C, as shown on the figure. We draw the
altitude from point
C, and call
H its intersection with the side
AB. The new triangle
ACH is
similar to our triangle
ABC, because they both have a right angle (by definition of the altitude), and they share the angle at
A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to
ABC. The similarities lead to the two ratios:
As
so
These can be written as
Summing these two equalities, we obtain
In other words, the Pythagorean theorem:
