## A Right Angled Triangle With Sides in Arithmetic Progression

The 3-4-5 triangle is familiar to anyone who has studies Pythagoras Theorem, This triangle is the smallest right angled triangle. The sides are in an arithmetic progression with first term 3 and common difference 1. Are there any other right angled triangles with sides in an arithmetic progression?
$x+2+(x+k)^2=(x+2k)^2$

$x^2+x^2_2kx+k^2=x^2+4kx+4k^2$

$x^2-2kx-3k^2=0$

$(x+k)(x-3k)=0$

Hence
$x+k=0 \rightarrow x=-k$
or
$x-3k=0 \rightarrow x=3k$
.
This means that the sides of the triangle are
$3k, \:4k, \: 5k$
- the triangle is just a scaled up 3-4-5 triangle.