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Prove that a triangle ABC is equilateral if and only if tanA+tanB+tanC=3underroot3....

Prove that a triangle ABC is equilateral if and only if


tanA+tanB+tanC=3underroot3....

Grade:12

2 Answers

Pratik Tibrewal
askIITians Faculty 37 Points
10 years ago
Using A.M. > = G.M.
(tanA + tanB + tanC)/3 >= (tanA.tanB.tanC)^(1/3)
=> tanA + tanB + tanC >= 3*(tanA + tanB + tanC)^(1/3)...
(since in triangle sum of tangents of angles is equal to product of tangents)
=> tanA + tanB + tanC >= 3(3)^(1/2)
tanA + tanB + tanC = 3(3)^(1/2) implies
A.M. = G.M. and this means tanA = tanB = tanC
=> A =B =C and hence triangle is equilateral,

If we take triangle as equilateral;
then all the angles are 60 degree, then tanA+ tanB + tanC = 3(3)^(1/2)

Thanks and Regards,
Pratik Tibrewal
askiitians faculty,
BTech IITG
Biswatosh Purkayastha
26 Points
7 years ago
Great... can be directly proved by the Jensen’s inequiality 
 
1/n[ f(x1)+f(x2)...f(xn) >= f[1/n(x1+x2+x3+......xn]
The equality holds good only if x1=x2=x3=....xn 
 

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