I'm not really sure if I got this, but I chose a 5,12,13 triangle. It satisfies the inequalities, but i'm not sure if they want a more general answer.Suppose that $\displaystyle a,b,c \in \mathbb{R}$ satisfy:

$\displaystyle 0<a<b+c \ \ \ \ \ 0<b<a+c \ \ \ \ \ 0<c<a+b$

i). Give a geometric construction to prove there exists a triangle in the Euclidean plane $\displaystyle \mathbb{E}^2$ (that is $\displaystyle \mathbb{R}^2$ with the usual metric) with sides of length equal to $\displaystyle a,b,c$.

With equality (I chose b=c), we get $\displaystyle 0<a<2c$ and $\displaystyle 0<c<a+c$. The second inequality seems a bit redundant since $\displaystyle a>0$.ii). What special thing happens in your construction if one of the inequalities becomes equality, say, $\displaystyle a=b+c$?

I don't really see the special thing that happens. My 5,12,13 triangle still works!