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Prove that a triangle A B C is equilater...

Prove that a triangle `A B C` is equilateral if and only if `tanA+tanB+tanC=3sqrt(3)dot`

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If the triangle is equilateral, then
`A=BC=60^(@)`
`rArr tan A+tan B+tan C=3 tan 60^(@)=3sqrt(3)`
Conversely assume that,
`tanA+tanB+tanC=3sqrt(3)`
But in `Delta ABC, A+B=180^(@)-C`
Taking tan on both sides we ge
`tan(A+B)=tan(180^(@)-C)`
`rArr (tan A+tanB)/(1-tanA tanB)=-tan C`
`rArr tanA+tanB=-tan C+tanAtanBtanC`
`rArr tanA+tanB+tan C=tanAtanBtanC=3sqrt(3)`
`rArr` None of the tan A, tan B, tan C can be negative
Also, `AMgeGM`
`rArr(1)/(3)[tan A+tan B+tan C] ge(tan A tan B tan C]^(1//3)`
`rArr(1)/(3)=(3sqrt(3))ge(3sqrt(3))^(1//3)rArr sqrt(3)ge sqrt(3)`
So, equality can hoold if and only if
tan A = tan B= tan C
or A=B=C or when the triangle is equilateral.
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