In , is the arithmetic mean and is the geometric mean of two positive numbers, then find minimum value of
If angles are opposites to side
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since the traingle is an issoceles, it has angles ( A , 2 π − 2 A , 2 π − 2 A ) . from the constraints of the problem: a = 2 k 1 + k 2 , b = c = k 1 k 2 . we use sin ( B ) = sin ( 2 π − 2 A ) = cos ( A / 2 ) and same for C, and sin ( 2 B ) = sin ( π − A ) = sin ( A ) to reduce the expression to 4 4 ( 2 sin ( A ) cos ( A ) + 2 sin ( A ) ) 4 2 cos 1 2 ( A / 2 ) + 1 = sin 4 ( A ) ( 2 cos ( A ) + 1 ) 4 2 cos 1 2 ( A / 2 ) + 1 = sin 4 ( A ) cos 8 ( A / 2 ) 2 cos 1 2 ( A / 2 ) + 1 ( 2 sin ( A / 2 ) cos ( A / 2 ) ) 4 2 cos 4 ( A / 2 ) + 1 = 8 sin 4 ( A / 2 ) 1 + 1 various double and half angle identity was used above. by laws of cosine, for an isosceles triangle, we know 2 b sin ( A / 2 ) = a → sin ( A / 2 ) = 2 b a = 4 k 1 k 2 k 1 + k 2 plugging this back in : 8 sin 4 ( A / 2 ) 1 + 1 = 3 2 ( k 1 + k 2 ) 4 k 1 2 k 2 2 + 1 = 3 2 ( 1 + k 1 / k 2 ) 4 ( k 1 / k 2 ) 2 + 1 we can minimize the function by minimizing 3 2 ( 1 + r ) 4 r 2 + 1 → d r d ( 1 + r ) 4 r 2 = 0 → ( 1 + r ) 4 2 r − ( 1 + r ) 5 4 r 2 = 0 → 2 r ( 1 + r ) − 4 r 2 = 2 r − 2 r 2 = 0 → r = 1 , 0 its obvious that 0 is not the answer, hence using one we have the expression equal 3 this is reached if k 1 = k 2