SOLUTION: 1. In a Triangle ABC, if cotA+cotB+cotC=√3 then prove that the triangle is equilateral triangle.
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Question 1017353: 1. In a Triangle ABC, if cotA+cotB+cotC=√3 then prove that the triangle is equilateral triangle.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
We use the identity that if A, B, C are angles of a triangle, then 
Dividing both sides by 
, we obtain
(1)
The left-hand side equals ^2 - \cot^2 A - \cot^2 B - \cot^2 C \right))
, so after some rearranging we get:
^2 = 2 + \cot^2 A + \cot^2 B + \cot^2 C)
Using the fact that 
:
(2)
(2)*2 + (1)*-2 yields the nice expression:
^2 + (\cot B - \cot C)^2 + (\cot C - \cot A)^2 = 0)
which implies that 
or 
. Therefore, 
.
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