Question

If the sides of a triangle are in Geometric Progression and the largest angle is twice the smallest angle then find the relation for $r$.

Answer 1

Let the sides of $\Delta$ be

$a,b=ar,c=a{r}^2$ where $r>1$

Here $C=2A$

So, $B=\pi -A-C-\pi -3A$

$\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

$\Rightarrow \frac{1}{\sin A}=\frac{r}{\sin B}=\frac{r^2}{\sin C}$

$\Rightarrow \frac{1}{\sin A}=\frac{r}{\sin 3A}=\frac{r^2}{\sin 2A}$

$\Rightarrow r^2=\frac{\sin 2A}{\sin A}=\frac{2\sin A\cos A}{\sin A}=2\cos A$

and $r=\frac{\sin 3A}{\sin A}=\frac{3\sin A-4{\sin }^{3}A}{\sin A}=3-4{\sin }^{2}A$

$\Rightarrow r=3-4(1-{\cos }^{2}A)=4{\cos }^{2}A-1$

$\therefore r^4-r-1=0$ is the required relation.