Question

In a triangle ABC, $r_1 \cot \dfrac{A}{2} + r_2 \cot \dfrac{B}{2} + r_3 \cot \dfrac{C}{2} = $ ?

• A. $s$
• B. $2s$
• C. $3s$
• D. $\dfrac{s}{2}$

Hint:

Each $r_i \cot\left(\dfrac{\text{angle}}{2}\right) = s$ in a triangle; so the sum over all three equals $3s$.

Answer 1

The Correct Option is C

Using the identity: \[ r_a \cot\left(\frac{A}{2}\right) = s,\quad r_b \cot\left(\frac{B}{2}\right) = s,\quad r_c \cot\left(\frac{C}{2}\right) = s \Rightarrow r_1 \cot \frac{A}{2} + r_2 \cot \frac{B}{2} + r_3 \cot \frac{C}{2} = s + s + s = 3s \]