Question: Reproduce a well-known result?

A classic result states that the equation x³＋px²＋qx＋r＝0 with real coefficients p, q, r has positive roots iff p＜0, q＞0, r＜0, and -27r² - 2p(2p² - 9q)r + q²(p² - 4q) ⩾ 0 (see for example this question). 
However, Maple appears unable to find the condition: 

a, b, c := allvalues(RootOf(x^3 + p*x^2 + q*x + r, x), 'implicit'):
RealDomain:-solve({a, b, c} >~ 0, [p, q, r]);
 = 
Warning, solutions may have been lost
                               []

Is there a way to get the above conditions in Maple with as little human intervention as possible (I mean, without a priori knowledge of the theory of polynomials)? 

Edit. An interesting problem is when these three positive roots can further be the lengths of sides of a triangle. For reference, here are some (unenlightening) results from some other software: