Question

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Answer 1

given cotA + cotB + cotC = sqrt3
to prove triangle ABC is equilateral 
we prove this by assuming ABC to be equialteral and establishing the truth 
of the statement cotA + cotB + cotC =sqrt3 
since ABC is equialteral angleA=angleB=angleC=60 degrees 
cotA=cotB=CotC = cot60= 1/sqrt3 
therefore cotA + cotB + cotC = 1/sqrt3 + 1/sqrt3 + 1/sqrt3 
=3/sqrt3 
=sqrt3 which is equal to the RHS ( right hand side) of the expression 
hence our assumption that ABC is equilateral is true 

Answer 2

let us assume to the contradictory that ∆ ABC is an equilateral triangle,

then Cot A = Cot B = Cot C =60°

=> (1/√3) + (1/√3) + (1/√3) = 3/√3 = √3

therefore L.H.S. = R.H.S

Hence our contradiction is true and ∆ ABC is equilateral.

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