Property of Triangle Altitudes

Given the sides b and c of a triangle . The third side is equal to the altitude drawn to it . Under this condition connecting b and c the triangle exists ,

$\frac{\sqrt{a} + b_{1}}{c_{1}} \leq \frac{b}{c} \leq \frac{\sqrt{d} + e}{f}$

Find $a + b_{1} + c_{1} + d + e + f.$

Here $a + b_{1} + c_{1} + d + e + f$ are integers. The expressions are in lowest terms.