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 satisfy:
i). Give a geometric construction to prove there exists a triangle in the Euclidean plane (that is with the usual metric) with sides of length equal to .
With equality (I chose b=c), we get and . The second inequality seems a bit redundant since .ii). What special thing happens in your construction if one of the inequalities becomes equality, say, ?
I don't really see the special thing that happens. My 5,12,13 triangle still works!