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If-a-b-c-0-prove-that-i-a-b-c-b-c-a-c-a-b-b-c-a-c-a-b-a-b-c-9-ii-a-b-c-b-c-a-c-a-b-b-c-a-c-a-b-a-b-c-9-

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Question Number 34357 by math1967 last updated on 05/May/18
If a+b+c=0 prove that i)((a/(b+c))+ (b/(c+a)) +(c/(a+b)))(((b+c)/a) +((c+a)/b) +((a+b)/c))=9 ii)((a/(b−c)) +(b/(c−a)) +(c/(a−b)))(((b−c)/a) +((c−a)/b) +((a−b)/c))=9
Ifa+b+c=0provethati)(ab+c+bc+a+ca+b)(b+ca+c+ab+a+bc)=9ii)(ab−c+bc−a+ca−b)(b−ca+c−ab+a−bc)=9
Answered by tanmay.chaudhury50@gmail.com last updated on 05/May/18
i){((a/(b+c))+1−1)+((b/(c+a))+1−1)+((c/(a+b))+1−1)}× {(((b+c)/a)+1−1)+(((c+a)/b)+1−1)+(((a+b)/c)+1−1)} ={(((a+b+c)/(b+c))−1)+(((a+b+c)/(c+a))−1)+(((a+b+c)/(a+b))−1)}× {(((a+b+c)/a)−1)+(((a+b+c)/b)−1)+(((a+b+c)/c)−1)} =given a+b+c=0 so =(−3)×(−3)=9
i){(ab+c+1−1)+(bc+a+1−1)+(ca+b+1−1)}×{(b+ca+1−1)+(c+ab+1−1)+(a+bc+1−1)}={(a+b+cb+c−1)+(a+b+cc+a−1)+(a+b+ca+b−1)}×{(a+b+ca−1)+(a+b+cb−1)+(a+b+cc−1)}=givena+b+c=0so=(−3)×(−3)=9

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