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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Similar questions

. If $a, b, c$ are the sides of a non-equilateral triangle, then the expression $(b + c - a) (c + a - b) (a + b - c) - a b c$ is:

a, b, c

\frac{a}{b + c - a} + \frac{b}{c + a - b} + \frac{c}{a + b - c}

a, b, c

a \neq 1, b + c \neq 1, c - b \neq 1

{log}_{b + c} a + {log}_{c - b} a = 2 {log}_{c + b} a {log}_{c - b} a

a

b

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∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣

a, b

c

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

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