Friday, April 1, 2011

Pythagorean Theorem

I am a little hesitant to post about a subject (the Pythagorean theorem), which is so well-known and covered so thoroughly on the web. Bronowski calls this relationship the "the most important single theorem in the whole of mathematics." (Ascent of Man: Ch. 4, Music of the Spheres). And I find these geometric proofs really wonderful to behold.

The graphic shows a right triangle and the squares of the sides in what is sometimes called the "Bride's Chair" form.

Let's start with the end rather than the beginning, and consider the last step. Drop an altitude in the right triangle as indicated by the dotted line. The argument will demonstrate that the two shaded areas in the figure (plum and red) are equal. That is the crux of the proof.

Then, by considering the symmetry of the figure, it is easy to see that the whole area of the red square is equal to the sum of the other two squares.

Now go back one step to the middle of the proof. The insight of the mathematician (who is really unknown) was to consider two special triangles as shown in the second figure (ADE on the left and ABG on the right).

Start with ADE. We take AE as the base of the triangle. Because the vertex D is on a line extended through FG, the height of the triangle is equal to EF, and therefore its area is one-half the area of the plum-colored square in the first figure.

Now look at ABG in the right panel. The altitude we dropped crosses AD at point H (marked as a red dot), and the shaded area we want is that part of the red square to the left of the dotted line, the area of rectangle ABIH. Take AB as the base of the triangle ABG. Then its height is equal to AH, and therefore its area is one-half the area of ABHI.

The first part of the proof is to show that these two carefully chosen triangles, ADE and ABG, are congruent (equal). The reason is SAS (side-angle-side). The side part is obvious. To get the angle, notice that ABG is just ADE rotated clockwise 90 degrees. Or, let the angle GAD = θ. Then, angle EAD = 90° + θ, but angle GAB = θ + 90°, which is the same.

This first proof illustrates a point about mathematical and scientific insight---its stepwise or incremental nature. We reach the summit by finding the crucial intermediate hand- and toe-holds.

There are many, many proofs of the Pythagorean theorem (e.g. here).

My absolute favorites are the geometric ones (#3 and #9), which are so simple that they really require no words. Here is #9, redrawn:

And finally, we can do it algebraically, although it feels a bit like cheating. Using the figure above, label the sides of the right triangle as a, b and the hypotenuse h. Then the sides of the large square on the left are a + b and its area is:

(a + b)2 = a2 + 2ab + b2

while the individual pieces are:

  = 4(1/2)ab + h2

Cancel and obtain:

a2 + b2 = h2

For a second method, drop the altitude h in the right triangle below, to divide the hypotenuse c into d + e. Notice that the two smaller triangles are similar to the original large one.

By similar triangles:

c/a = a/d
cd = a2

c/b = b/e
ce = b2

a2 + b2 = cd + ce
= c(d + e) = c2