There is a triangle that is not equilateral whose side lengths form a geometric sequence, and the measures of whose angles form an arithmetic sequence. Show that this statement is true by finding such a triangle or prove that it is false by demonstrating that there cannot be such a triangle.
There is already a question on this site regarding this problem but I want someone to review my solution because it is different from the one that already exists on this site. This question came in Euclid Math Contest, $2022$.
Let the sides of the triangle be $a,ar,ar^2$ then the angle opposite to $ar^2$ will be $60+d$, angle opposite to $ar$ will be $60$ and angle opposite to $a$ will be $60-d$ where $d,r>0$.
So now by cosine law, $$(ar)^2=a^2+(ar^2)^2-2(a)(ar^2)\cos60$$ $$\implies r^4-2r^2+1=0$$ $$\implies r=\pm1$$ But $r>0$ and $r\ne1$ $$\boxed{\implies\textrm{No triangle exist}}$$
Any help is greatly appreciated.