Sides of a Right Angled Triangle With Sides in a Geometric Progression

Suppose we have a right angled triangle with sides in a geometric progression. The sides must be of the form  
\[a, \: ar, \: ar^2\]
. Suppose  
\[r \gt 1\]
  then  
\[ar^2\]
  is the longest side, the hypotenuse, and we can use Pythagoras Theorem to write
\[a^2+(ar)^2=(ar^2)^2\]

\[a^2+a^2r^2=a^2r^4\]

\[1+r^2=r^4\]

\[r^4-r^2-1=0\]

Then  
\[r^2=\frac{1 \pm \sqrt{(-1)^2 - 4 \times 1 \times (-1)}}{2 \times 1} = \frac{1 \pm \sqrt{5}}{2}\]

Since  
\[r \gt 1\]
, the solution is  
\[r^2=\frac{1+ \sqrt{5}}{2} \rightarrow r=\sqrt{\frac{1+\sqrt{5}}{2}}\]