Suppose that the three quadratic equations ax² –2bx+c = 0, bx² –2cx+a = 0 and cx² –2ax+b = 0 all have only positive roots. Then

Since the roots of the three equations are positive,
all of a, b, c have the same sign.
Also ${\mathrm{b}}^2\ge \mathrm{ac},{\mathrm{c}}^2\ge \mathrm{ab}$ and a² $\ge$ bc .
Hence ${\mathrm{b}}^2{\mathrm{c}}^2\ge {\mathrm{a}}^2\left(\mathrm{bc}\right)$  which implies $bc\ge a^2$
Therefore a² = bc similarly b² = ca and c² = ab.
Thus a = b = c.