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Show that for any triangle with sides `a ,b ,a n dc3(a b+b c+c a)<(a+b+c)^2<4(b c+c a+a b)dot` When are the first two expressions equal ?

Text Solution

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By using triangular inequlaity,
`clta+b`
`rArr c^(2)ltca+ab`
Similarly, `a^(2)ab+ac and b^(2)lt bc+ab`
`:. a^(2)+b^(2)+c^(2)2ab+2bc+2ca`
`rArr (a^(2)+b^(2)+c^(2))+2ab+2bc+2ca+lt4(ab+bc+ca)`
`rArr (a+b+c)^(2)lt4 (ab+bc+ca" "....(i)`
Now, using AM-GM inequalioty in ab,b and c, we get
`(a^(2)+b^(1))/(2)ge ab,(b^(2)+c^(2))/(2)ge bc and (c^(2)+a^(2))/(2) geca`
`rArr a^(2)+b^(2)+c^(2)ge ab+bc+ca`
`rArr a^(2)+b^(2)c^(2)2ab+2bc+2cage2(ab+bc+ca)`
`rArr (a+b+c)^(2)ge3(ab+bc+ca)" "...(ii)`
From Eqa. (i) and (ii), we get
`3(ab+bc+ca)le(a+b+c)^(2)le4(ab+bc+c)`
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