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  
\[r \gt 1\]
  is the longest side, the hypotenuse, and we can use Pythagoras Theorem to write




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

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