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Question

In a Δ ABC, if B=60, prove that (a+b+c)(ab+c)=3ca

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Solution

Now, It is given that :(a+b+c)(ab+c)=3ca a2ab+ac+abb2+bc+acbc+c2=3ca a2b2+c2+ac=3ca a2b2+c2=2ca a2b22ca=b2 ....(i)We have, B=60cos B=12 a2+b2b22ac=12 a2+c2b2=ac a2+c2ac=b2 ....(ii)(a+b+c)(ab+c)=3caa2ab+ac+abb2+bc+acbc+c2=3ac=a2+c2b2+2ac3ac=0a2+c2ac=b2 ....(iii)from (i)~(ii)~and~(iii)If B=60 and (a+b+c)=3ca hence proved.


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