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First consider : -
( a cotA + b cotB + c cotC ) / 3 R
2R (cosA + cosB + cosC ) / 3 R
cosA + cosB + cosC 3 / 2
2 cos(A+B)/2 cos(A-B)/2 + 1 - 2 sin2C/2 3 / 2
This a quadratic equation in sinC/2 withnegative coefficient of sin2C/2
D = b2 - 4ac
= 4 cos2(A - B)/2 - 4
since cos is always less than 0 therefore the eqauation is always valid with a critical point
Therefore,
since r = 4RsinA/2sinB/2SinC/2
or to prove that
On solving,
Consider s-b, s-c and use AM GM
similiarly for sinB/2 and sinC/2
therefore,
sinA/2sinB/2SinC/2 1/8
let sinA/2sinB/2SinC/2 = k
To prove 4k ( 4k + 1) / 3
or, 12k 4k + 1
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