Animated Proof of Pythagoras' Theorem

Introduction

Everyone knows Pythagoras's Theorem:

"the area of the square on the hypotenuse of a right angled triangle
is equal to the sum of the areas of the squares on the 2 shorter sides".

But many of us were convinced of it as children only by long-winded, complicated, and unsatisfying proofs.

This page presents an elegant and simple proof which relies only on 2 very simple lemmas,
each of which is easily demonstrated visually.

No algebra, trigonometry, or calculation of any sort is required.

Lemmas

Lemma 1:

If 2 triangles share 2 angles and the length of an equivalent side (with respect to the angles),
the triangles are congruent (meaning "identical or mirror-images").

Informal proof: Two angles being the same makes the triangles similar
(the same shape), fixing the length of a side makes them congruent.

Lemma 2:

If a parallogram is distorted by moving one side along its extension
(ie in a direction parallel to its opposite side, a distortion we call a
"parallelogram shear"), then the area of the parallelogram is not
altered. (if you're not sure what this means, go look at the proof animation
and it'll probably become clear).

Informal proof: This lemma can be shown to be a corollary of lemma 1,
which guarantees that when performing a shear, the triangle that is added
at one end is congruent to the triangle that is removed at the other end.
Of course, lemma 2 also follows from the formula for the area of a parallelogram:
Area = length of side x perpendicular distance from the opposite side,
since a parallelogram shear leaves the 2 values on the right hand side
of this formula unchanged.

Click here to see the proof.

Links

One of Dudeney's proofs: An animated proof using Java, but requiring some maths (not given).
Pythagoras site page: Presents the same proof, as I was taught it at school. Without the
animation it's not so obvious that the bits fit together. This site has some
dodgy javascript but seems to work ok (just close or press ok on the error
messages as they appear). It has some other stuff like...
President Garfield's proof: Algebra!
Another proof: More algebra.
Award-winning Euclid's proof: This won a Sun Java programming contest in 1995. It's very similar to the
one here but uses triangles instead of parallelograms, and they have to be rotated as well as distorted twice.
The School of Pythagoras: Errrr.... "Sacred Geometry".
Living Maths Project: More mathsy Java applets.