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Prove that a triangle ABC is equilateral if and only if tan A + tan B + tan C = 3√3.

Prove that a triangle ABC is equilateral if and only if
tan A + tan B + tan C = 3√3. 

Grade:11

1 Answers

Jitender Pal
askIITians Faculty 365 Points
9 years ago
Hello Student,
Please find the answer to your question
Let ABC is an equilateral ∆ then
A = B = C = 60°
⇒ tan A + tan B tan C = 3√5
Conversely, suppose
tan A + tan B + tan C = 3√3 . . . . . . . . . . (1)
Now using A. M. ≥ G. M. (equality occurs when no’s are equal)
For tan A, tan B, tan C, we get
tan A + tan B + tan C/3 ≥ (tan A tan B tan C)1/3
NOTE THIS STEP :
But in any ∆ABC, know that
tan A + tan B tan C = tan A tan B tan C
∴ Last inequality becomes
tan A + tan B + tan C/3 ≥(tan A + tan B + tan C)1/3
⇒ (tan A + tan B + tan C)2/3 ≥ 3
⇒ tan A + tan B + tan C ≥ 3√3
Where equality occurs when tan A, tan B, tan C are equal, i.e. A = B = C
⇒ ∆ABC is equilateral.

Thanks
Jitender Pal
askIITians Faculty

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