Problem #WSP-000028

Problem

There is a triangle with side lengths $a$, $b$ and $c$. Can you form a triangle with side lengths $\frac{a}{b}$, $\frac{b}{c}$ and $\frac{c}{a}$? Does it depend on what $a$, $b$ and $c$ are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on $a$, $b$ and $c$.
Recall that a triangle can be drawn with side lengths $x$, $y$ and $z$ if and only if $x+y>z$, $y+z>x$ and $z+x>y$.