The Pythagorean Theorem is perhaps the most famous theorem in geometry, if not in all of mathematics. In this lab, we look at one method of proving the Pythagorean Theorem by constructing a special square. Part I of this report describes the construction used in the proof and Part II gives a detailed explanation of why this construction works, that is why the construction generates a proof of the Pythagorean theorem. Finally, we conclude with some comments on the many proofs of the Pythagorean Theorem.

To start out our investigation of the Pythagorean Theorem, we assume that we have a right triangle with legs b and a and hypotenuse c. Our first task construction is that of a segment sub-divided into two parts of lengths a and b. Since a and b are arbitrary, we just create a segment, attach a point, hide the original segment, and draw two new segments as shown.

Then, we construct a square on side a and a square on side b. The purpose of doing this is to create two regions whose total area is a^2 + b^2. Clever huh? Constructing the squares involved several rotations, but was otherwise straightforward.

The next construction was a bit tricky. We define a translation from B to A and translate point C to get point H. Then, we connect H to D and H to G, resulting in two right triangles. In part II, we will prove that both of these right triangles are congruent to the original right triangle.

Next, we hide segment BC and create segments BH and HC. This is so that we have well-defined triangle sides for the next step - rotating right triangle ADH 90 degrees about its top vertex, and right triangle HGC -90 degrees about its top vertex.

We will now prove that this construction yields a square (on DH) of side length c, and thus, since the area of this square is clearly equal to the sum of the areas of the original two squares, we have a^2+b^2 = c^2, and our proof would be complete. By SAS, triangle HCG must be congruent to the original right triangle, and thus its hypotenuse must be c. Also, by SAS, triangle DAH is also congruent to the original triangle, and so its hypotenuse is also c. Then, angles AHD and CHG(= ADH) must sum to 90 degrees, and the angle DHG is a right angle. Thus, we have shown that the construction yields a square on DH of side length c, and our proof is complete.

This was a very elegant proof of the Pythagorean Theorem. In researching the topic of proofs of the Pythagorean Theorem, we discovered that over 300 proofs of this theorem have been discovered. Elisha Scott Loomis, a mathematics teacher from Ohio, compiled many of these proofs into a book titled The Pythagorean Proposition, published in 1928. This tidbit of historical lore was gleaned from the Ask Dr. Math website (http://mathforum.org/library/drmath/view/62539.html). It seems that people cannot get enough of proofs of the Pythagorean Theorem.