Property of Triangle Altitudes

Geometry Level 4

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.

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