Pythagorean Triples

Pythagorean Triples are sets of three integers for which the statement, A² + B² = C², is true. This is a special case of the Pythagorean Theorem which states for all right triangles, A² + B² = C². While the most common introduction of the Pythagorean Theorem to students uses the triangle with sides, A=3, B=4, C=5, right triangles are not limited to sides of integer length. The Pythagorean Triple for this triangle is (3,4,5). An example of a right triangle with non-integers sides is represented by the triangle form by the two sides and the diagonal of an 8.5 x 11 inch sheet of paper. This triangle has sides of 8.5, 11 and 13.901 inches (13.901 is the square root of the sum of 8.5 square and 11 square).

Other Pythagorean triples include (5,12,13), (8,15,17) and (7,24,25). There are an infinite number of Pythagorean Triples.

I recently came across a formula to generate additional Triples.

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

For example, starting with n=2 and m=1, one generates the (3.4,5) Triple.

2nm = 4
n² - m² = 3
n² +m² = 5

Another example, with n=3, n=2, generates the (5,12,13) Triple

2nm = 12
n² - m² = 5
n² +m² = 13

I wrote an Excel spreadsheet to generate Triples up to n=12, m=11 and produced the following:

N	M		A	B	C
2	1		3	4	5
3	1		8	6	10
3	2		5	12	13
4	1		15	8	17
4	2		12	16	20
4	3		7	24	25
5	1		24	10	26
5	2		21	20	29
5	3		16	30	34
5	4		9	40	41
6	1		35	12	37
6	2		32	24	40
6	3		27	36	45
6	4		20	48	52
6	5		11	60	61
7	1		48	14	50
7	2		45	28	53
7	3		40	42	58
7	4		33	56	65
7	5		24	70	74
7	6		13	84	85
8	1		63	16	65
8	2		60	32	68
8	3		55	48	73
8	4		48	64	80
8	5		39	80	89
8	6		28	96	100
8	7		15	112	113
9	1		80	18	82
9	2		77	36	85
9	3		72	54	90
9	4		65	72	97
9	5		56	90	106
9	6		45	108	117
9	7		32	126	130
9	8		17	144	145
10	1		99	20	101
10	2		96	40	104
10	3		91	60	109
10	4		84	80	116
10	5		75	100	125
10	6		64	120	136
10	7		51	140	149
10	8		36	160	164
10	9		19	180	181
11	1		120	22	122
11	2		117	44	125
11	3		112	66	130
11	4		105	88	137
11	5		96	110	146
11	6		85	132	157
11	7		72	154	170
11	8		57	176	185
11	9		40	198	202
11	10		21	220	221
12	1		143	24	145
12	2		140	48	148
12	3		135	72	153
12	4		128	96	160
12	5		119	120	169
12	6		108	144	180
12	7		95	168	193
12	8		80	192	208
12	9		63	216	225
12	10		44	240	244
12	11		23	264	265

In the table above, while C is always the longest leg, either A or B can be the shortest side of the triangle.

Also, certain Triples are highlighted yellow. These Triples are unique in that they produce triangles that are not similar to triangles produced by any prior set of Triples. For example, the set (12,16,20) has sides all divisible by 4 so it produces a triangle that is similar to (3,4,5). While the set (8,15,17) is highlighted in that the three numbers has greatest common denominator (GCD) of 1. Therefore, its triangle is not similar to the triangle of any prior set of Triples.

Generator:
Pythagorean Triple Generator
Update 9/24/2022 - see post Math Vacation: Pythagorean Triple Generators - Part III (jamesmacmath.blogspot.com) for additional methods of producing Pythagorean Triples. The generator spreadsheet (above) has been updated with these additional methods.

Pythagorean Presidential Smarts

So this blog isn't wasn't you are expecting if you "hate" or "love" our president. 144 years ago, James Garfield developed a proof to the Pythagorean Theorem. A few years later, he was elected the 20th President of the United States of America in 1880. 

There are many proofs to the Pythagorean Theorem. Garfield's proof, while similar to some prior proofs, is very concise and easy to understand.

Take any right-angle triangle (ABC) and make a duplicate triangle (A'AC') that is rotated and adjacent to it that you arrange as shown:

Add a line connecting points B and A'.

Now, name the sides of each triangle for the point at its opposite angle. Note, the the length A' = A and C' = C:

Garfield calculated the area of the trapezoid two ways. First, by addition of parts - adding the area of the three triangles is C²/2 + AB/2 + (AB)/2 =  C²/2 + AB

The second way to calculated the area of the trapezoid is the using the formula, base times 1/2 the sum of the two sides, or: (A + B) x 1/2 x (A + B).= A²/2 + B²/2 + AB

Since both calculated areas represent the same, C²/2 + AB = A²/2 + B²/2 + AB

Subtract AB from both sides of the equation and multiply both sides by 2, the result is:

C² = A² + B²

Inscribed Angle Theorem of Thales

The original theorem, as stated by the Greek mathematician Thales, is all triangles inscribed in a circle with two corners diametrically opposed are right triangles. Note: right triangles are triangles with one angle that is 90 degrees (a right angle).

Lesson

First look at some of the example triangles inscribed in the circles below. They all appear to be right triangles.

Proof

Although all the examples drawn above have right angles (the angles with yellow square), that doesn’t prove that all such inscribed triangles should be right triangles.

The proof of Thales uses three known facts that were previously established by early mathematicians. Fact 1 is that all points on a circle are the same distance from the center of the circle (this is actually the definition of a circle). Fact 2 is that any triangle with two equal sides (an isosceles triangle) also has two equal angles. Fact 3 is that the sum of the interior angles of a triangle equals 180 degrees (this is also another lesson in this book). Let’s draw an inscribed triangle within the circle and we’ll label the angles of the triangle as A, B, C.

For the next step of the proof, we’ll add a new point, X, on the diameter at the center of the circle. Draw a line from X to the corner at angle C (shown above).  Also, the angle C is now split into two angles which are labelled A* and B*. 

We’ll use the three given facts from above. Using Fact 1, the length of line XC equals the length of line XA. Using Fact 2, the triangle AXC has two sides of equal length therefore the angle A* is the same as the angle A.

Repeating these steps for the BXC; using Fact 1 the lengths of lines XB and XC are the same. Using Fact 2, angle B* is the same as angle B.

Finally using Fact 3, the angles A + B + C = 180 degrees. Angle C equals A* + B* which is the same as A+B.  Substituting C for A+B in Fact 3 gives us C + C = 180 degrees or 2C= 180 degrees. Therefore angle C = 180/2 = 90 degrees (or angle C is a right angle).

Alternate Application

The theorem can also be used in reverse to locate the center of any circle. Start with any circle and place a sheet of rectangular paper (any size will do) so a corner is touching the circle (see point A). Mark the points where the two sides coming off that corner intersect the circle (see points B and C). Draw a line between points B and C – these points are now diametrically opposed.

Update 8/23/2021:

Some believe Thales should be given credit for first visualizing a proof to the Pythagorean Theorem: See: Should we rename the Pythagorean theorem? - Big Think.