Conceptually, one of the most attractive ways to' prove Pythagoras' Theorem is to find a partition the three squares into smaller regions, with the property that the partition of the large square is assembled from regions in the partitions of the two other squares. Such proofs are among the oldest known. Some of them turn out to be closely related to the straightforward `algebraic' proofs.

There is in fact a general and elementary result about areas in the plane (apparently proven first around 1900) which asserts that if two polygonal figures in the plane have the same area, then one can find piecewise congruent decomposition of the two figures. That is to say, on grounds of general principle one knows that if Pythagoras' Theorem is true, one can find the sort of partitions we are looking for.

David Hilbert, **Foundations of Geometry**, Open House, 1994.

On each side
we have a square of side *a + b*.
On the left inscribed in it is a
square of side *c*, and four copies
of the original right triangle.
The area of the square on the left is therefore

On the right we have the two smaller squares of Pythagoras' Theorem and the same four triangles. The area is therefore

If we subtract the area of the four triangles from each of these expressions, we get

The algebra here asserts that

*
c ^{2} = 4(ab/2) + (b-a)^{2}
= a^{2} + b^{2}*.

Tufte's book
(**Envisioning Information**, p. 84)
has a representation
in the colours attributed to the figure in about 200 A.D.
(This is as far as I know the earliest known use of colour in
mathematical diagrams.)

This proof also gives rise implicitly to a dissection proof, attributed (see Heath, volume I, p.64) to Thabit ibn Qurra from about 900 A.D.

Here is a version of this explained by Michael Buckland, an undergraduate, to a Geometry course at UBC:

*
Click in control node to adjust size and reset the figure.
Click anywhere else to start and stop the animation.
*

**Reference**
Jean Claude Martzloff,
**A history of Chinese mathematics**,
Springer 1997.

**Reference:**

* ... in Mr. Perigal's elegant proof ... the four parts into which he
has divided the greater side are equal in all respects, so that the
division of the squares is symmetrical.
In this consists chiefly the elegance of
the construction, which is in this respect ... unique.*
J. W. L. Glaisher in an addendum to Perigal's note.

*Both coloured nodes move.*

**References:**

Felix Bernstein,
`Der Pythagoraische Lehrsatz',
*Zeitschriften fur Mathematischen und Naturwissenschaftlichen Unterricht *55 (1924),
pages 204 - 207.

K. O. Friedrichs, *From Pythagoras to Einstein*,
M. A. A. New Mathematical Library,
pages 8 - 12.

Major P. MacMahon, `Pythagoras's Theorem as a repeating pattern', Nature 109 (1922), page 479.

Paul Mahlo, `Topologische Untersuchungen uber Zerlegung in ebene und spharische Polygone', Dissertation, Halle, 1908.

Greg N. Frederickson,
*Dissections:
Plane & Fancy*,
Cambridge University Press, 1997, pages 28 - 31.
(I wish to thank Frederickson for pointing out the work
of Mahlo, MacMahon, and Bernstein.)

One way to decompose the large square into two regions, each one
matching areas with one of the smaller squares.
Of course the rectangles in the large square
are not themselves squares.
According to the principle
asserted at the beginning of this page, however,
one can decompose each of these rectangles
and each of the smaller squares into matching congruent regions.
Describe how to do this. *This is not so easy,
because the number of regions required will depend
on the exact triangle one starts with*.
It will help to refer to
the discussion of shears, where a
similar question is answered for parallelograms
with the same base but different heights.

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