# Pythagorean Triangle/Examples/1380-19,019-19,069

## Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $1380$, $19 \, 019$ and $19 \, 069$ is a primitive Pythagorean triangle.

File:1380-19,019-19,069.png

It has generator $\left({138, 5}\right)$.

## Proof

We have:

 $\ds 138^2 - 5^2$ $=$ $\ds 19 \, 044 - 25$ $\ds$ $=$ $\ds 19 \, 019$

 $\ds 2 \times 138 \times 5$ $=$ $\ds 1380$

 $\ds 138^2 + 5^2$ $=$ $\ds 19 \, 044 + 25$ $\ds$ $=$ $\ds 19 \, 069$

 $\ds 1380^2 + 19 \, 019^2$ $=$ $\ds 1 \, 904 \, 400 + 361 \, 722 \, 361$ $\ds$ $=$ $\ds 363 \, 626 \, 761$ $\ds$ $=$ $\ds 19 \, 069^2$

It follows by Pythagoras's Theorem that $1380$, $19 \, 019$ and $19 \, 069$ form a Pythagorean triple.

We have that:

 $\ds 1380$ $=$ $\ds 2^2 \times 3 \times 5 \times 23$ $\ds 19 \, 019$ $=$ $\ds 7 \times 11 \times 13 \times 19$

It is seen that $1380$ and $19 \, 019$ share no prime factors.

That is, $1380$ and $19 \, 019$ are coprime.

Hence, by definition, $1380$, $19 \, 019$ and $19 \, 069$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$