Question 96996
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