In triangle ABC, prove that the maximum value of `tanA/2tanB/2tanC/2i s R/(2s)`

Explanation

To prove that the maximum value of \(\frac{\tan A/2 \tan B/2 \tan C/2}{R/(2s)}\) in triangle ABC is \(\frac{R}{2s}\), we will follow these steps:

### Step 1: Write the expressions for \(\tan A/2\), \(\tan B/2\), and \(\tan C/2\)

Using the half-angle tangent formulas, we have:
\[ \tan \frac{A}{2} = \sqrt{\frac{s - a}{s}}, \quad \tan \frac{B}{2} = \sqrt{\frac{s - b}{s}}, \quad \tan \frac{C}{2} = \sqrt{\frac{s - c}{s}} \]
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