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.
