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E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b} , right triangle, inradii

Source: 2010 Moldova JBMO TST p6

February 25, 2021

geometryalgebraGeometric Inequalitiesinequalities

Problem Statement

In a right triangle with the length legs

b

and

c

, and the length hypotenuse

a

, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed

1 + \sqrt2

. Determine the numerical value of the expression of

E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}