Explanation of Mathematics

      A mathematics teacher from Ohio, Elisha Scott Loomis, spent his entire life collecting proofs of the Pythagorean Theorem. He included 371 of them in his book, The Pythagorean Proposition. He received hundreds more proofs after the book was released, but he couldn’t keep up with the collection. No one really knows how many distinct proofs exist of the Pythagorean Theorem, especially since so many are similar in many ways.

Even though we think of the Pythagorean theorem as an algebraic equation, a2+b2=c2, Pythagoras considered it a geometric statement about area. Formally the theorem is stated in terms of area as well. “In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).” It is usually summarized as follows: “The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.”

Some famous people have studied the theorem and have come up with their own proofs as well. President James Garfield developed one around 1876, about 5 years before he became president. It was later published in the New England Journal of Education. His proof dealt with a trapezoid and the areas of 3 triangles. His proof follows:

First, we need to find the area of the trapezoid by using the area formula of the trapezoid: A=½h(b1+b2)

In the diagram on the right, h=a+b, b1=a, and b2=b.

A=½(a+b)(a+b) =½(a2+2ab+b2)

Now, let's find the area of the trapezoid by summing the area of the three right triangles.

The area of the red triangle is A=½(ba). The area of the green triangle is A=½(c2).

The area of the blue triangle is A= ½(ab).

The sum of the area of the triangles is ½(ba) + ½(c2) + ½(ab) = ½(ba + c2 + ab) = ½(2ab + c2).

Since, this area is equal to the area of the trapezoid we have the following relation:

½(a2 + 2ab + b2) = ½(2ab + c2).

Multiplying both sides by 2 and subtracting 2ab from both sides we get

a2+b2=c2



Einstein's Proof of the Pythagorean Theorem

Einstein was another who claimed to have proved the Pythagorean Theorem at the young age of 12. Although he left no evidence of this proof, but described it in general terms, mentioning that it relied on “the similarity of triangles.” He also said that it took “much effort.” Twenty-five years ago, a physicist named Manfred Schroeder presented a proof that he believed was how Einstein had proved the Pythagorean Theorem. In his book “Fractals, Chaos, Power Laws,” Schroeder wrote that a friend of his had showed him this proof. That friend had heard it from one of Einstein’s former assistants, who had heard it from Einstein himself. We can never be sure if this is how Einstein proved the Pythagorean Theorem, but it shows all the characteristics of Einstein’s work. A key element to understanding the proof is to understand that right triangles can be decomposed into two smaller copies of itself. That can only be done with right triangles. So Einstein's proof proves that the Pythagorean theorem only applies to right triangles. It is a fairly simple proof as follows:

Step 1: Draw a perpendicular line from the hypotenuse to the right angle. This partitions the original right triangle into two smaller right triangles.

Step 2: Note that the area of the little triangle plus the area of the medium triangle equals the area of the big triangle.

Step 3: The three triangles are similar because their corresponding angles are equal, and their corresponding sides are in proportion.

Step 4: Because the triangles are similar, each occupies the same fraction f of the area of the square on its hypotenuse. Restated symbolically, this observation says that the triangles have areas fa2, fb2, and fc2, as indicated in the diagram.

Step 5: Remember, from Step 2, that the little and medium triangles add up to the original big one. Hence, from Step 4, fa2 + fb2 = fc2.

Step 6: Divide both sides of the equation above by f. You will obtain a2 + b2 = c2, which says that the areas of the squares add up. That’s the Pythagorean theorem.

The Bride's Chair proof

From Euclid's Elements

One of the most famous proofs is one commonly called the Bride's Chair proof, the Peacock's tail proof, or the windmill proof. Euclid wanted to show that the area of the smaller squares equaled the area of the larger square. I've created an interactive applet to guide you through a simple version of this proof. If you'd like to use it in a larger window, click here.

A simple visual representation of how the area of the squares on the sides of a right triangle is equal to the area of the square on the hypotenuse. (Courtesy Mike Jacobs on YouTube https://www.youtube.com/watch?v=Ohg-Cww2gN0 )

An interesting point about the pythagoras theorem is that you could actually choose any shape to lie on the lengths of the triangle – it does NOT have to be a square. It just has to be similar and the same shape. Alex Bellos notes in his fabulous book, “Alex´s Adventures in Numberland,” that we could even use a Mona Lisa if we wanted to. Maybe at this point you´re starting to see that pythagoras is just as much about area as it is about length, possibly even more! Explore this applet to see for yourself. Open in a new window here.