The next result is attributed to Archimedes: The equation $x^3-ax^2+(4/9)a^2b$ has a positive root if and only if $a>3b$, where $a$ and $b$ are positive real numbers.
This problem appeared in the book The Heritage of Thales by W.S. Anglin and J. Lambek. The idea was proved with the tools of the time.
I tried to interpret the equivalent equation $\Big(\frac{2a}{3x}\Big)^2=\frac{a-x}{b}$ to show through proportions what you want: area, segments, etc. But I can not find a way to solve it. Is it the way to proceed? Any suggestions?
P.S.: Sorry for my English, I'm not native. :v