Exploring Proofs of Pythagorean Theorem

By Rishan Bheda

Since its discovery, the Pythagorean Theorem has remained the cornerstone of the mathematical realm and propelled advancements and understanding within it. As stated by German astronomer and mathematician, Johannes Kepler in his cosmological work ‘Mysterium Cosmographicum,’ “Geometry has two great treasures: One is the Theorem of Pythagoras and the other division of line into mean and extreme ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” 

Even though archaeological evidence has proved that Egyptians, Babylonians, and Chinese people discovered the theorem before him, the theorem is generally credited to Pythagoras, a Greek mathematician. This is the case as he was the first to bring widespread awareness to this theorem. The Pythagorean Theorem boasts many practical applications that have contributed significantly to developing tools and machinery that have helped shape our modern world. The theorem also serves as a basis for numerous other mathematical laws. The Pythagorean Theorem has been the subject of various studies and about 400 different proofs of the theorem are known today.  In the following paper, I will be exploring some of the proof, in particular two new proofs that were recently discovered.

In Euclidian geometry, the Pythagorean theorem states that: If the two legs of a right triangle have the lengths a and b, while c is the length of the hypotenuse, the side opposing the right angle, of that triangle then the sum of the two areas of the squares on the legs is equivalent to the area of the square on the hypotenuse.

This leads to the infamous equation:

a2+b2=c2

In the following work, I explore a proof technique of this Pythagorean theorem that makes use of four congruent framing triangles, as shown in Figure 1, to illustrate the derivation of the theorem using a diagram. The small frame model and the large frame model are the two standard framing models as shown in Figure 2 and Figure 3 (see endnotes).

 

Brief History of the Theorem

The principles that make up this theorem, although named after Pythagoras, can be traced all the way back to before the ancient Greeks. Many historians suggest that the ancient Babylonians possessed knowledge of this fundamental right triangular relationship. A major artefact that supports this notion is the Plimpton 322 tablet, dating back to around 1800 BCE, which features meticulous records of various Pythagorean triples, presenting integer solutions of the equation a2+b2=c2. Similarly, the Yale YBC 7289 tablet, also from around the same period, showcases a square with a side length of thirty, accompanied by an incredibly precise approximation of 2, conveyed through Babylonian sexagesimal numerals. The Pythagoreans, followers of the teachings of Pythagoras, are believed to eventually develop this theorem further in the 6th century BCE. 

 

History of Proofs through Frames

The framing proofs highlighted in the previous sections unveil captivating historical ties. Notably, the amalgamation of the "large and small frame" diagram, as depicted in Figure 5, traces its origins back to the ancient Chinese text known as Zhoubi Suanjing believed to have been written between 100 BCE and 100 CE. Although primarily an astronomy text, the Zhoubi Suanjing acknowledges this principle as the Gougu rule. Proof number 253 in The Pythagorean Proposition demonstrates how the Pythagorean relationship can be affirmed through a similar diagram. Furthermore, the 12th-century Indian mathematician Bhaskara incorporated a diagram akin to the ancient Chinese version in his renowned work Lilavati further reinforcing the cross-cultural transmission and adaptation of mathematical concepts across civilizations.

 

Another notable framing proof employs half of the large frame diagram arranged in a trapezoidal configuration, as shown in Figure 5. In 1876, prior to assuming office as the twentieth President of the United States, James Garfield, then an Ohio Representative, utilized a diagram akin to Figure 6 to establish the Pythagorean theorem. This finding was subsequently published in the New England Journal of Education. 

Throughout history, numerous scholars have derived the Pythagorean theorem through various methods. It is important to mention that pinpointing the origins of these proofs is a rather imprecise science. While we may possess a general understanding of how different concepts evolved and who contributed to them, in certain instances, it can be arduous or nearly impossible to definitively ascertain the exact origins of a particular proof.

Some Proof Techniques

The Pythagorean Theorem offers multiple derivations, each categorized by its distinct proof techniques. This section provides a concise overview of the dissection and similar triangle proof methods before delving into the workings of the large and small frame proofs. 

1. Dissection Proofs

Dissection proofs involve sophisticated arguments that deconstruct the square on the hypotenuse into regions that can be rearranged into the two squares on the legs. Euclid's renowned windmill diagram, depicted in Figure 6, serves as an exemplary demonstration of such proof. Another elegant dissection argument, attributed to the 19th-century amateur British mathematician Henry Perigal, utilizes the diagram showcased in Figure 7. This diagram is dissected using vertical and horizontal lines from the midpoint of each side of the hypotenuse square, along with lines passing through the centre of the longer leg's square, which are parallel and perpendicular to the hypotenuse. Further details regarding this derivation can be found in Casselman's article on Henry Perigal and his proof.

2. Small and Large Frame Proof

For the large frame proof, we use a standard right triangle, like the one shown in Figure 1, to build the pair of diagrams shown in Figure 8 using two sets of four congruent copies of the triangle.

Through the equal area argument, these two diagrams provide a visual demonstration of the fact that a2+b2=c2. Using a more algebraic approach we can equate the total area of the frame on the left with the sum of its five parts.
The area of the frame comes to be (a+b)2, while the area of each triangle is  12ab. This leads to

a+b2=412ab+c2a2+2ab+b2=2ab+c2a2+b2=c2

For the small frame proof, we once again make use of two sets of four congruent right triangles to build the pair of diagrams shown in Figure 9.

These two diagrams provide another proof for the Pythagorean identity using the equal area argument. Once again this can be presented using algebra. We have

c2=412ab+b-a2

                                         =2ab+b2-2ab+a2

                                         =b2+a2

 

3. Similar Triangle Proof

Another type of proof of the theorem makes use of similar triangles. The diagram shown in Figure 9 is one form of this proof in which a larger right triangle is divided into two smaller right triangles by its altitude. 

 

Two New Frames

The search for new ways to look at proving this theorem is ongoing. After some experimentation with different arrangements, two new arrangements to prove the Pythagorean theorem were discovered by two math professors, Ian Adelstein and George Ashline, named the ‘overlapping frame’ and ‘ninja star frame.’ Both these arrangements seamlessly fit back into the original small frame arrangement.

a. Ninja Star Frame

To achieve the frame outlined in this proof, we overlap the initial four congruent triangles, but this time forming a ninja star frame as shown in Figure 10. This arrangement cuts the triangles along their altitude leading to triangles that are similar to the original ones. Due to the fact that the green triangles form a small frame, we use an argument similar to the one used before to equate the area of the frame with the sum of the areas of the four congruent triangles and the square in the centre of the frame.

b. Overlapping Frame

Once more, we examine four identical red replicas of the initial right triangle depicted in Figure 1. This time, we position them within a frame that intersects the four triangles, as illustrated in Figure 11. This layout divides each of the original red triangles along its altitude, generating groups of triangles that are similar to the initial ones. Among these sets of similar triangles, represented by blue in Figure 11, one adopts the small frame arrangement as seen in Figure 3. Consequently, the blue triangle fulfils the Pythagorean identity, and by virtue of similarity, the original red triangle does as well.

To prove this algebraically we denote the lengths of the red triangle with a, b, and c and the corresponding lengths of the blue triangle with x, y, and b. As the blue and red triangles are similar, we get

ax=by=cb and a2x2=b2y2=c2b2

Using these inequalities, we can derive that

c2=c2b2*b2=c2b2*x2+y2=c2b2*x2+c2b2*y2=a2x2*x2+b2y2*y2=a2+b2

 

 

 

References

N. Beryozin, The good old Pythagorean theorem, Quantum, Jan./Feb. 1994, 25-28, 60-61. Adelstein, I. M., & Ashline, G. L. (2019). Reframing the Pythagorean Theorem. The College Mathematics Journal, 50(1), 28–35. https://doi.org/10.1080/07468342.2019.1557983

Maor, E. (2007). The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press.

Bogolmony, A. (2012). Pythagorean theorem and its many proofs, interactive mathematics miscellany and puzzles. https://www.cut-the-knot.org/pythagoras/index.shtml.

Kepler, J. (1596). Mysterium Cosmographicum de Admirabile Proportine Orbium Celestium