Note that the figure below shows a large rectange divided into two halves of equal area by the thick horizontal line. At the center of the figure is the right triangle around which the proof revolves - the square of the length of each side of the right triangle is represented as a square attached to the triangle; note the labels for a2, b2, and c2.
In the course of the proof we subtract out equal pieces of area one at a time from each side of the thick line, until all that remains at the end are the three squares, with a2 and b2 on one side of the thick line and c2 on the other. Since we originally began with equal halves and we only removed equivalent areas from each side, these remaining areas must be equivalent. The colors show what congruent areas cancel out of each side, leaving the three squares at the end.
What remains is that we show that at the beginning the two halves are congruent, and that each of the equivalently-colored shapes inside are congruent. This is simply done using geometric comparisons such as side-side-angle, side-side-side, and so on, of the triangles. (Remember those?) The two yellow rectangles get divided up into triangles for this as well.
a2 + b2 =
c2
Lastly, here are some related
webpages you might find interesting:
- Mathworld's
Pythagorean Theorem page, which contains several more
proofs.
- The website for a PBS Nova
episode about Fermat's Last Theorem which relates
closely to Pythagorus' theorem. This site includes a
fascinating story about the famous woman mathematician
Sophie Germain and how she worked on Fermat's Last
theorem with Gauss.
- Wikipedia's
Pythagorean Theorem page, which has some really neat
history, and some generalizations of the theory regarding
vectors and non-Euclidean geometry. (If you don't know
what those things mean, don't let that stop you from
reading the history part or the geometric proofs....)