Prove the following :
$r_{1} r_{2} r_{3} = r^{3}$ $c o t^{2}$ (A/2) $c o t^{2}$ (B/2) $c o t^{2}$ (C/2)

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Solution

To prove: $r_{1} r_{2} r_{3} = r^{2} {cot}^{2} \frac{A}{2} {cot}^{2} \frac{B}{2} {cot}^{2} \frac{C}{2}$
We know,
$\sqrt{r r_{1} r_{2} r_{3}} = r^{2} cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2} ⟹ r r_{1} r_{2} r_{3} = r^{4} {cot}^{2} \frac{A}{2} {cot}^{2} \frac{B}{2} {cot}^{2} \frac{C}{2} ⟹ r_{1} r_{2} r_{3} = r^{3} {cot}^{2} \frac{A}{2} {cot}^{2} \frac{B}{2} {cot}^{2} \frac{C}{2}$

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