SEMESTER PROJECT


EXPLANATION OF MATHEMATICS



There are over 300 proofs of the Pythagorean theorem. Mathematics teacher Elisha Scott Loomis (1852-1940) spent his life writing 371 proofs in his book The Pythagorean Proposition, published in 1927 (Ratner, 2009). Others who have created their own proofs of the theorem include Albert Einstein, Euclid, Leonardo da Vinci, and James Garfield. Multiple videos and applets have been made on the subject as well. Consider this applet by the user Skoledu which demonstrates two different geometric proofs.



GeoGebra Applet by user Skoledu

Albert Einstein wrote a proof of the Pythagorean theorem based on similar triangles and their properties. He was twelve years old at the time, and his life was changed because of the theorem. Years later, he used it in a fourth-dimensional form in the Special Theory of Relativity and in the General Theory of Relativity, where Einstein applied the Pythagorean theorem to three coordinates and set them equal to the displacement of a ray of light (Ratner, 2009).


Euclid's Elements includes two proofs of the theorem, the first being Proposition 47 of Book I and the second being Proposition 31 of Book VI. The former relies on area relations, while the latter is based on proportion. Proposition 31 of Book VI states, ''In right-angled triangles, the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.'' Proposition 47 of Book I states, ''In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.'' The latter is much simpler and is one of the most popular proofs of the Pythagorean theorem. This proof has many names, including but not limited to, ''the windmill,'' ''the bride's chair,'' ''franciscan's cowl,'' and ''the peacock's tail'' (Bogomolny, 2003).



Figure 4: Sketch of Proposition 47 of Book I. The sum of the squares containing the right angle is equal to the square opposite the right angle (Bogomolny, 2016).


Proposition 31 of Book VI from Euclid's Elements provides another proof of the Pythagorean theorem. Because this proof is more complex than the proof Euclid's other proposition, consider viewing Proof #7 from Pythagorean Theorem and its Many Proofs by Alexander Bogomolny. This website also contains a collection of 117 more proofs of the Pythagorean theorem, if the reader is interested in learning more.




Figure 5: Euclid's proof of the converse of the Pythagorean theorem


Along with the two proofs of the Pythagorean theorem, Euclid provided a proof for the converse of the theorem, which is used to test if a triangle is a right triangle. Consider Figure 5 above. Given a triangle △ ABC, let a2 + b2 = c2, so (AB)2+(AC)2=(BC)2. Then, draw the segment AD, which is equal to AB, perpendicular to AC. This forms the right angle DAC. Connect point D to point C, creating the segment CD. Since AD=AB, (AD)2=(AB)2. Then, adding (AC)2 to both sides, we get


(AD)2+(AC)2=(AB)2+(AC)2.


Now, because angle DAC is a right angle, we know that


(AD)2+(AC)2=(CD)2.


We also know that


(AB)2+(AC)2=(BC)2,


since that was given. Putting these two facts together gives


(CD)2=(BC)2,


so CD=BC. Now, because AD=AB and the segment AC is shared between the two triangles, AD=AB, AC=AC, and CD=BC, so angle DAC is equal to angle BAC. Since angle DAC is a right angle, this also means that angle BAC is a right angle. Therefore, the triangle is a right triangle (Joyce, 1996).




Figure 6: Pythagorean theorem proof by James Garfield


United States President James Garfield was interested in the Pythagorean theorem and made a proof of his own utilizing the area of a trapezoid. As shown in Figure 6 above, the area is given by the sum of the three triangles, so Area=\(\frac{1}{2}\) ab+\(\frac{1}{2}\) ab+\(\frac{1}{2}\) c2=ab+\(\frac{1}{2}\) c2. Additionally, the area of the entire trapezoid is shown by Area=\(\frac{1}{2}\)(height)(sum of bases) =\(\frac{1}{2}\)(a+b)(a+b)=\(\frac{1}{2}\)(a2+2ab+b2). Then, because the areas are equal, we get ab+\(\frac{1}{2}\) c2=\(\frac{1}{2}\) (a2+2ab+b2). Simplifying this equation further gives a2+b2=c2, the Pythagorean theorem (Gaskins, 2008).


At ten years old, Andrew Wiles used textbook methods to try to prove Fermat's Last Theorem. This theorem states that there are no natural numbers x,y, and z such that xn+yn=zn in which n is a natural number greater than 2 (Britannica). This theorem can be seen as the generalization of the Pythagorean theorem. Later, Wiles studied the work of mathematicians who tried to prove it, but he decided to leave Fermat's Last Theorem to study elliptic curves in his graduate studies. His new study in elliptic curves provided a path to prove Fermat's Last Theorem. Ratner (2009) wrote, ''Andrew Wiles' most famous mathematical result is that all rational semi-stable elliptic curves are modular, which, in particular, implies Fermat's Last Theorem.'' His proof took about seven years to complete, and in turn made him famous.


Mathematicians across the globe have been fascinated by Pythagoras and the theorem named after him. For hundreds of years, humans have been developing proof after proof to show that a2+b2=c2. One common proof uses areas of the squares of the right triangle's legs to show that the sum of those squares is equal to square of the hypotenuse. This proof has been duplicated several times in different ways. Consider this original Geogebra applet which shows another way the areas of the squares of the legs can be used to express the area of the hypotenuse.



Original GeoGebra Applet by Tyra Ah Kuoi
(In order for the applet to work, a must be less than b. However, if a>b in context, the triangle can be reflected so that the longer side is along the bottom.)