BALKAN-SL 2020 A4

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$ , $\rho_2$ , $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$ , $C(\rho_2, 0)$ , $D(\rho_3, 0)$ . If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$ , find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.

Brazitikos Silouanos, Greece