Pythagorean Theorem: Subtle Dangers of Visual Proof

This is a dynamic illustration of a faulty proof of the Pythagorean Theorem. The argument is by dissection and rearrangement and is included as geometric proof 22 in Loomis' collection.

The proof resembles others of this variety: for example, proofs 26, 27, 36. However, unlike those mentioned, this proof contains a subtle point that was apparently overlooked by E. S. Loomis. The construction as it is described in the book can't be carried through in some of the possible configurations of the Bride's Chair. To see the reason, drag the right-angled vertex to make one of the legs considerably shorter than the other.

This is now listed as #3 among the faulty proofs of the Pythagorean theorem. The applet above shows that the argument can be amended into a real proof by considering the dynamics of the dissections as has been done, e.g. when comparing by equidecomposition two parallelograms with the same altitude and base.

References

1. E. S. Loomis, The Pythagorean Proposition, NCTM, 1968