Proofs of the Pythagorean Theorem

Kemunto Ondande, Alex Nelson, and Hunter Brehm

Part A Report on at least three other proofs of the

Pythagorean theorem besides the ones illustrated in

this chapter. If you find further interesting historical

information about this great theorem, report on that

too.

Garfield

● James A. Garfield was the twentieth president of the United States.

● Before becoming president, he was a professor at Western Reserve Eclectic

Institute, now called Hiram College, in Hiram, Ohio.

● In addition to being a mathematician, he had unusual abilities such as being

able to write Greek in the left hand while writing Latin in the right hand.

● Although he was aware of many proofs of the Pythagorean Theorem, he was

able to construct his own unique proof.

● He developed his proof in 1876 while working as a Member of Congress.

Garfield’s Proof

His proof consists of a diagram of a

trapezoid with a parallel sides a and b and

height a + b. Garfield looked at the area of

the diagram in two different ways: as that of

a trapezoid and as that of three triangles,

two of which are congruent.

Garfield’s Proof

Proof:

● Start with a right triangle with legs a and b and hypotenuse c.

● Extend leg a by b units and construct a duplicate right triangle along this

extension.

● Upper leg a is parallel to the original leg b since, in a plane, if a line is

perpendicular to each of two lines, then the two lines are parallel.

● Draw a segment XY to close the figure.

● The resulting quadrilateral is a trapezoid with bases a and b and altitude a + b.

● The trapezoid is composed of two congruent right triangles and right triangle

XYZ. Triangle XYZ is isosceles since two of its sides have length c. Angle

XZY is a right angle since ∡ 1 + ∡ XZY + ∡ 2 = 180˚ and ∡ 1 + ∡ 2 = 90˚.

● Therefore, the area of the trapezoid equals the sum of three right triangles of

which it is composed, two of which are congruent.

Garfield’s Proof

Algebraic expression:

● Area of a trapezoid = 1

2(sum of the parallel sides)(the height of the

trapezoid)

● 𝐴 =1

2(𝑎 + 𝑏)(𝑎 + 𝑏)

● Area of the scalene right triangle: 𝐴𝑠 =1

2𝑎𝑏

● Area of the isosceles right triangle: 𝐴𝑖 =1

2𝑐2

● 𝐴𝑡 = 2 𝐴𝑠 + 𝐴𝑖

● 𝐴 =1

2𝑎 + 𝑏 𝑎 + 𝑏 = 2

1

2𝑎𝑏 +

1

2𝑐2

● 𝑎 + 𝑏 𝑎 + 𝑏 = 2𝑎𝑏 + 𝑐2

● 𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐2

● 𝑎2 + 𝑏2 = 𝑐2

Bhaskara

Bhaskara was an Indian mathematician in the 12th century that proved the Pythagorean Theorem.

He developed two proofs of the Pythagorean Theorem:

1st Proof: consists of placing congruent right triangles around a small square in such a way that the

hypotenuses of each triangle form a larger square on the outside

2nd Proof: consists of drawing an altitude of a right triangle to hypotenuse c and then using similarity

between the triangle and the two smaller triangles within it, formed by the altitude, to prove the

theorem

Bhaskara’s 1st Proof of Pythagorean Theorem

● 𝐴 = 𝑐2

● Area of the blue triangles = 41

2𝑎𝑏 = 2𝑎𝑏

● Area of the yellow square = (𝑏 − 𝑎)2

● Area of the big square = 41

2𝑎𝑏 + (𝑏 − 𝑎)2

= 2ab + 𝑏2 − 2𝑎𝑏 + 𝑎2

= 𝑎2 + 𝑏2

𝑐2 = 𝑎2 + 𝑏2

Bhaskara’s 2nd Proof of Pythagorean Theorem

Proof:

● Draw an altitude on the hypotenuse of a right triangle

● Show that the right triangle & one of the two congruent triangles that the

altitude formed were similar.

● We will let r be 𝑐1 for triangle ACE and let s be 𝑐2 for triangle CBE.

● Prove the triangle ABC ~ triangle CBE

● Show triangle ABC ~ triangle ACE

● Add the two similar triangles: 𝑠𝑐 + 𝑟𝑐 = 𝑎2 + 𝑏2

● Since we know that 𝑐 = 𝑠 + 𝑟. We can solve this algebraically.

● So, 𝑠𝑐 + 𝑟𝑐 = 𝑎2 + 𝑏2

𝑐 𝑠 + 𝑟 = 𝑎2 + 𝑏2

𝑐2 = 𝑎2 + 𝑏2

Part BA Pythagorean triple (a, b, c) of positive integers

satisfying the Pythagorean equation. The triple is

primitive if the integers have no common factor. A

general Pythagorean triple is a positive integer

multiple of a primitive one (cancel the gcd). Find the

polynomials p, q, r of degree 2 in two integer

variables such that every primitive Pythagorean

triple is given by a = p(m, n), b = q(m, n), and c =

r(m, n) and conversely these equations provide a

primitive Pythagorean triple for every pair of

unequal relatively prime positive integers (m, n).

Search the web for further results on Pythagorean

triples and report on the results you find most

interesting.

Polynomials for Primitive Pythagorean Triple

● The formula for the Pythagorean Theorem can be easily converted into the equation for the unit circle.

Pythagorean Theorem:

Further Results on Pythagorean Triples

● Many years ago before the Pythagoras, the Babylonians knew how to create Pythagorean triples using natural

numbers: (a,b,c) such that 𝑎2 + 𝑏2 = 𝑐2.

● If (m,n) is a pair of natural numbers with m > n, then: (a,b,c) = (𝑚2 − 𝑛2, 2𝑚𝑛,𝑚2 + 𝑛2) is a Pythagorean triple.

[(m,n) is consider the generator of (a,b,c)]

● If (a,b,c) is any Pythagorean triple and d is any natural number, then (da,db,dc) is consider a Pythagorean triple.

● Another way in generating primitive Pythagorean triples is by the following: 𝑎, 𝑏, 𝑐 = (𝑟𝑠,𝑟2−𝑠2

2,𝑟2+𝑠2

2) where (r, s)

is a pair of odd integers , greater common denominator (gcd) of (r, s)=1, and r > s ≥ 1

● A solution to the Diophantine equation 𝑥2 + 𝑦2 = 𝑧2 (Pythagorean triples)

● There are no Pythagorean triples in which both legs of a triangle are odd.

○ Theorem: If (a,b,c) is a Pythagorean triple, then at least one of the positive integers a or b is even.

References

Ballard, William R. Geometry. Philadelphia, Saunders, 1970.

Bogomolny, Alexander. Pythagorean Theorem and Its Many Proofs, Alexander Bogomolny,

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Eckert, E. (1992). Primitive Pythagorean Triples. The College Mathematics Journal, 23(5), 413-417. doi:10.2307/2686417

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jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html.

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Garfield’s Proof of the Pythagorean Theorem | Mathematical Association of America, Mathematical Association of America, Feb.

2016, https://www.maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem

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