One of the most important and famous formulas in mathematics is the Pythagorean Theorem: for a right triangle, the square of the long side (hypotenuse) is equal to the sum of the squares of the other two sides. Using a diagram:

As I have pointed out in other posts, proofs of major results like this are usually left out of high school level texts, and some students would probably like to see where the formula comes from.

In the case of the Pythagorean Theorem, there are literally hundreds of proofs (Maor, 2007). In this post I will just show two. The first uses some algebra, and is one of the easiest to understand. The second is Euclid’s classic proof. The Euclid proof is a tour de force of geometrical logic, and is featured as one the classic math results in Durham’s book “Journey into Genius” (Durham, 1991).

First the algebra-based proof: take four identical right triangles, and arrange them as follows:

Now, we have an outer square of size a+b. This square is made up of four triangles, plus an inner square of side c.

In terms of areas:

(area of square a+b) = (4 x triangle area) + (area of square c), or

Clearing the parenthesis,

Eliminating 2ab from both sides,

That’s it!

Now I’ll move on to the second proof – the most famous, and the one Euclid himself discovered.

		Euclid’s proof is purely geometric. In this diagram, a square has been grafted onto each of the right triangle’s three sides, and the areas of the squares are a2, b2, and c2. Thus, we need to prove that the total area of the two smaller squares equals the area of the larger square.
		For the next step, Euclid drops a line from the right angle, perpendicular to the hypotenuse, dividing the large square into two parts, the blue rectangle, and the green rectangle. We are going to show that the blue rectangle has the same area as the blue square, and the green rectangle and the green square have the same area. Doing this will complete the proof, since the total area of the big square (blue plus green), will equal the area of the two smaller squares.
	Before we proceed, first remember a result from high school geometry: two triangles are identical (congruent) if two sides and the angle between them are equal. This is usually called the SAS (side-angle-side). The two triangles in the diagram at left are congruent if AB = XY, BC = YZ, and angle B = angle Y.
			In this diagram, I have drawn two more lines, forming triangles ABC and DBE. These two are congruent because: AB = DB (they are each sides of the same square) BE = BC (also sides of a common square) angle ABC = angle DBE (each angle equals 90 degrees plus the common angle DBC) Thus the triangles are congruent by SAS.

Now surprisingly, we are almost done:

Consider the area of triangle ABC. It equals ½ x AB x BD (1/2 base x height). In other words, it’s half the area of the small square, AB x BD

Consider the area of triangle DBE. It equals ½ x BE x EF. In other words, it’s half the area of the small rectangle, BE x EF.

So, one of the identical triangles has area of half the small square. The other identical triangle has area of half the small rectangle. Therefore, the square and rectangle have the same area!

We can pretty much stop here, because exactly the same reasoning can be used to show that the large rectangle has area equal to the upper right square.

In conclusion:

Area _{Blue Rectangle} = Area _{Blue Square}
Area _{Green Rectangle} = Area _{Green Square}

Adding these two equalities:

Area _{Blue Rectangle}+Area _{Green Rectangle}=Area _{Blue Square}+Area_{Green Square}

And we’re done.

The last diagram above is justly famous, and is sometimes called “Euclid’s Windmill”, because of its vague resemblance to a Dutch windmill. Supposedly, the great mathematician Karl Gauss once suggested that the diagram be laid out on a gigantic scale on some flat area on the earth as a signal that intelligent life exists here. Gauss thought any intelligent beings would inevitably discover the same proof, and would be familiar with the drawing.

Bibliography

Durham, W. (1991). Journey through Genius: The Great Theorems of Mathematics. Penguin.

Maor, E. (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton University Press.

Tags: Math

This entry was posted on Thursday, July 2nd, 2009 at 11:04 pm and is filed under Mathematics History, Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.