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Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+...

Prove that `(b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)gt a+b+c` , where `a,b,cgt 0`.

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`(b^(2) + c^(2))/(2) gt ((b + c)/(2))^(2)` or `(b^(2) + c^(2))/(b + c) gt (1)/(2) (b + c)`
Similarly,
`(a^(2) + b^(2))/(a + b) gt (1)/(2) (a + b)` and `(c^(2) + a^(2))/(c + a) gt (1)/(2) (c + a)`
Adding, we get
`(b^(2) + c^(2))/( b + c) + (c^(2) + a^(2))/(c + a) + (a^(2) + b^(2))/(a + b) gt a + b + c`
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