Pythagorean Triangles


Pythagorean Numbers

Pythagorean triangles are right triangles with integer numbers for lengths of sides. An example would be a triangle with sides 3, 4 and 5. This triangle is a right triangle since 3^2 + 4^2 = 5^2 or 9 + 16 = 25. You can tell we are using the "Pythagorean" idea that for sides a, b and c of a right triangle a^2 + b^2 = c^2. We will call a collection of three natural numbers a Pythagorean Triple if they represent lengths of sides of a Pythagorean triangle.

If {3, 4, 5} is a Pythagorean Triple, PT, then so is {6, 8, 10} and we will let you prove that. You may want to demonstrate that for any natural number, n, {na, nb, nc} is a Pythagorean Triple when {a, b, c} is a Pythagorean Triple. We will call {a, b, c} a Unique Pythagorean Triple, UPT, if a, b and c do not have a common factor n. This also means that Unique Pythagorean Triples have unique angles since we are eliminating similar triangles.

We can make Prime Pythagorean Triples using two numbers x and y. To accomplish this let a = 2xy, b = x^2 - y^2 and c = x^2 + y^2 where y is less than x. Now show that a^2 + b^2 = c^2. What about reversing the direction where you are given any Pythagorean Triple a, b and c then find an x and y that yields a = 2xy , b = x^2 - y^2 and c = x^2 + y^2 where y is less than x?

How can we pick x and y, with y less than x, in order that we get a Unique Pythagorean Triple? Try using natural numbers that have a greatest commom factor of one and opposite parity. In other words, x and y should be relatively prime and if one is odd the other is even. Of course part of the parity issue is taken care of since x even and y even implies a common factor of 2.

Write a short program for your computer that gives a list of Unique Pythagorean Triples. One outside loop can give natural numbers x and for each x an inside loop could give natural numbers y from one to the number x. For each x and y in this nested loop you can check to see if the greatest common factor is 1 and check to see if x and y have opposite parity. For opposite parity: Mod(x,2) not equal to Mod(y,2) would check the natural number remainder of x divided by 2 against the remainder of y divided by 2. Try creating a few UPTs by hand and with a program on your calculator or computer.

For any Pythagorean Triple {a, b, c} we can show that a/c, b/c and 1 satisfy the Pythagorean identity m^2 + n^2 = 1 and therefore (a/c, b/c) is a rational point on a unit circle.

Rational Points

Pythagorean Triples Slopes on a unit circle.
Finding Pythagorean Triples Lines through a unit circle.
Pythagorean Triples overlap parts 3, 4 and 5
Pythagorean Triples Indexed Into Series
Problems for math classes

For more information and ideas about working with Pythagorean Numbers you may link to Web pages from the following math artists class: Danielle, Darlene, Kellie, Jill, Nanyal, Kevin, Rachel, Christina, Jennifer, Candice, Vance, Esther and Melinda. Most of us are in this picture but not in order of names.

Another Project About Triangles

Salomon Mosseri, Email, edited this page.

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