Prove that $cos A + cos B + cos C = 1 + r / R$

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Solution

$cos A + cos B + cos C = 1 + 4 sin (\frac{A}{2}) sin (\frac{B}{2}) sin (\frac{C}{2})$
$= 1 + \frac{[4 R sin (A / 2) sin (B / 2) sin (C / 2)]}{R} = 1 + \frac{r}{R}$

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