Introduction
A Pythagorean triple is an ordered triplet of positive integers (a,b,c) such that:
a^2+b^2=c^2.
It is evident that such integers correspond to the sides of a right triangle, a, b being the catheti (legs) and c being the hypotenuse of the triangle. The most well known Pythagorean triple is 3,4,5.
As early as in the 8th grade I started feeling that knowing at least a few commonly used Pythagorean triples allows solving various geometry problems with a bigger ease. For example, it helps you spot right triangles and solve for the third side in a triangle. This proved to be a valuable skill when dealing with comparatively primitive geometry problems in elementary school.
In this exploration I will investigate two ways how a Pythagorean triple can be generated. First, the Euclid’s theorem of generating these triplets will be explored and proved that the values generated, with the help of this formula, are in fact a Pythagorean triple corresponding to sides of a right triangle. Next, the Berggren’s Parent/Child relationships will be explored.
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two methods of generating primitive Pythagorean triples were explored, such as, through the application of Euclid’s formula and through the application of Berggren’s Parent/Child Relationships. After having explored both methods, a way of generating not only primitive, but also non-primitive triples became evident. Furthermore, in this exploration I was able to prove that by using the Euclid’s formula it is possible to generate three values that correspond to the catheti and hypotenuse of a right triangle, hence all primitive Pythagorean triples. As a great advantage to this exploration I have gained a more thorough understanding of Pythagorean triples and how these can be generated, this topic can be easily linked to the 2D geometry course studied in