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Inequalities Involving Mathematical Means

Let a, b, c b...

Question

Let a, b, c be the sides of a triangle. The least value of expression $E = \frac{a}{b + c - a} + \frac{b}{c + a - b} + \frac{c}{a + b - c}$ is ___

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Solution

$P u t x = b + c - a, y = c + a - b, z = a + b - c, t h e n x, y, z > 0 a n d 2 a = y + z, 2 b = z + x, 2 c = x + y U s i n g A . M . \geq G . M ., w e g e t E = \frac{1}{2} (\frac{y + z}{x} + \frac{z + x}{y} + \frac{x + y}{z}) \geq 3 {(\frac{y}{x} . \frac{z}{x} . \frac{z}{y} . \frac{x}{y} . \frac{x}{z} . \frac{y}{z})}^{1 / 6} = 3$

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