given the following three distances $r_i, r_j, r_u$ and following relationship
$$0< \frac{r_u}{r_i} < 1$$ $$0< \frac{r_u}{r_j} < 1$$
What is the relation between $$\frac{r_i}{r_j} = ?$$
To further elaborate, I need the minimum of $\frac{r_u}{r_i}$ and $\frac{r_u}{r_i}$, so it means I should need the maximum of $\frac{r_i}{r_j}$. This I have found through multiple iterations in Matlab. But I need mathematical proof.
The image below shows the depiction of how my scenario looks like!
Try something like this......
$r_u + x = r_i$ where $0<x<\infty$ ........which satisfies $0<\frac{r_u}{r_i}<1$
Similarly $r_u + y = r_j$ where $0<y<\infty$ ......which satisfies $0<\frac{r_u}{r_j}<1$
Therefore $0<\frac{r_u+x}{r_u+y}<\infty$ ranging from $x\to0$ and $y\to\infty$ to
$x\to\infty$ and $y\to0$ where $r_u$ is a finite number
Hence $0<\frac{r_i}{r_j}<\infty$