Home: Pythagorean Theorem and Proof Essay

Professor R. Smullyan in his book 5000 B.C. and Other Philosophical Fantasies tells of an experiment he ran in one of his geometry classes. He drew a right triangle on the board with squares on the hypotenuse and legs and observed the fact the the square on the hypotenuse had a larger area than either of the other two squares. Then he asked, "Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which would you choose?" Interestingly enough, about half the class opted for the one large square and half for the two small squares. Both groups were equally amazed when told that it would make no difference.

The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares equals (the area of) the big one.

In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle.

The theorem is of fundamental importance in Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.

Below is a collection of 104 approaches to proving the theorem. Many of the proofs are accompanied by interactive Java illustrations.

Remark

The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that its converse is also true. The converse states that a triangle whose sides satisfy a² + b² = c² is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact.

W. Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968. In the Foreword, the author rightly asserts that the number of algebraic proofs is limitless as is also the number of geometric proofs, but that the proposition admits no trigonometric proof. Curiously, nowhere in the book does Loomis mention Euclid's VI.31 even when offering it and the variants as algebraic proofs 1 and 93 or as geometric proof 230.

In all likelihood, Loomis drew inspiration from a series of short articles in The American Mathematical Monthly published by B. F. Yanney and J. A. Calderhead in 1896-1899. Counting possible variations in calculations derived from the same geometric configurations, the potential number of proofs there grew into thousands. For example, the authors counted 45 proofs based on the diagram of proof #6 and virtually as many based on the diagram of #19 below. I'll give an example of their approach in proof #56. (In all, there were 100 "shorthand" proofs.)

I must admit that, concerning the existence of a trigonometric proof, I have been siding with with Elisha Loomis until very recently, i.e., until I was informed of Proof #84. Actually, for some people it came as a surprise that anybody could doubt the existence of trigonometric proofs, so more of them have eventaully found their way to these pages.

In trigonometric terms, the Pythagorean theorem asserts that in a triangle ABC, the equality sin²A + sin²B = 1 is equivalent to the angle at C being right. A more symmetric assertion is that ΔABC is right iff sin²A + sin²B + sin²C = 2. By the sine law, the latter is equivalent to a² + b² + c² = 2d², where d is the diameter of the circumcircle. Another form of the same property is cos²A + cos²B

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