The key fact about similarity is that **as a triangle scales,
the ratio of its sides remains constant**.

There is a very simple proof of Pythagoras' Theorem that uses the notion of similarity and some algebra. It is commonly seen in secondary school texts. We can cut the triangle into two parts by dropping a perpendicular onto the hypothenuse. Since these triangles and the original one have the same angles, all three are similar. Therefore

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x / a = a / c, c - x / b = b/c
x = a ^{2} / c, c - x = b^{2}/c
x + (c-x) = c, a^{2}/c + b^{2}/c = c
a^{2} + b^{2} = c^{2}*

This proof is not so far removed from Euclid's proof. That proof shows by a geometric argument that

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a ^{2} = x c
b^{2} = (c-x) c
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whereas this follows immediately from the equations above.

As the quotation from Einstein suggests, this is a natural line of reasoning to arrive at if one is told the statement of Pythagoras' Theorem. Heath suggests that this may have been the ``proof'' originally found by the Greeks, especially because (1) it is not so far from the proof in Euclid (which tradition asserts to be original with him), and (2) rather reminiscent of Euclid's proof of VI.31, a generalization of Pythagoras' Theorem whose proof depends on the theory of ratios and in effect gives a second proof of Pythagoras' Theorem. The main point is that a proof by similarity arguments depends on a good theory of ratios, which have only been developed in Euclid's Book V. The theory of ratios is complicated by the existence of incommensurable segments (such as the sides and diagonal of a square), and is one of the most sophisticated parts of all of Greek mathematics.

To a modern mind, this proof is attractive because it emphasizes that the Theorem is essentially a consequence of the scalability of Euclidean geometry, in other words a kind of geometrical recursion. Other proofs generally rely more directly on the properties of translation, which means eventually on the properties of parallelism, perhaps less immediately intuitive.

Similarity can also be used
in a geometrical argument.
Many proofs of Pythagoras' Theorem show that the
area of the square on the hypotenuse
is the sum of areas of the squares
constructed on the other sides.
It is not necessary to use squares for this argument to work.
It suffices to replace squares by any three similar figures
whose sizes are proportional to the three sides of
the triangle, since if *s* is the ratio of the
area of this figure to that of a square,
then the equations

*a ^{2} + b^{2} = c^{2}
*

s a^{2} + s b^{2} = s c^{2}

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are equivalent. The simplest thing to do is use
the triangles we constructed above. A complete figure for
this argument might look like this:
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Of course the area of the larger triangle
is the sum of the areas of the smaller triangles
precisely because we can
fold them back to get the second figure above.
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