Proving Euclid’s Formula Using Basic Geometry
Discussing Integer Solutions to x²+y²=z²
Natural number (positive integer) solutions to the equation x²+y²=z² are called Pythagorean Triples.
Example: x=3, y=4, z=5 is a Pythagorean Triple which is typically written in the short form (3, 4, 5).
We can obtain a Pythagorean Triple by using Euclid’s Formula. This says that (x, y, z) is a Pythagorean Triple if and only if x=2mn, y=m²-n² and z=m²+n², for some integers m and n where m>n>0.
Example: m=5 and n=1 will generate the Pythagorean Triple (10,24,26).
The ‘if’ part is easily verified since x²+y²=(2mn)²+(m²-n²)²=(m²+n²)²=z².
The ‘only if’ part is not so straightforward. In a previous piece, we discussed two methods for deriving Euclid’s formula.
In this piece, we discuss a third method using geometry where we look at points on a circle.
Euclid’s Formula Using Basic Geometry
If we plot the graph x²+y²=1, we get the unit circle i.e. a circle with radius 1, whose origin is at (0,0):
