Question

In \(\triangle ABC\), \(r r_1 \cot^ \frac{A}{2} + r r_2 \cot^ \frac{B}{2} + r r_3 \cot^ \frac{C}{2} = \)

• A. \( 3\Delta \)
• B. \( 3S \)
• C. \( \frac{S}{\Delta} \)
• D. \( \Delta \)

Hint:

In problems involving the inradius, exradii, and cotangents, recognize how these quantities relate to the area of the triangle. Identifying geometric identities or known formulas can simplify complex trigonometric expressions.

Answer 1

The Correct Option is A

Step 1: Establish the relevance of the product \( r r_i \) in the triangle's geometry, where \( r \) is the inradius and \( r_i \) are the exradii of the triangle, each associated with an angle \( A, B, \) or \( C \) respectively. These radii have a deep connection to the area of the triangle. 
Step 2: Recognize that \( \cot \frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2} \) are related to the semiperimeter \( s \) and the side lengths of the triangle. Specifically, for each angle, we can express \( \cot \frac{A}{2} \) as: \[ \cot \frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \] Similarly for \( \cot \frac{B}{2} \) and \( \cot \frac{C}{2} \), and their products with the exradii \( r_1, r_2, r_3 \). 
Step 3: The given expression sums the products of \( r r_i \) and the cotangents of the half-angles. Using known geometric identities, this expression simplifies to a multiple of the area \( \Delta \) of the triangle: \[ r r_1 \cot^ \frac{A}{2} + r r_2 \cot^ \frac{B}{2} + r r_3 \cot^ \frac{C}{2} = 3\Delta \] where \( \Delta \) is the area of the triangle. 
Step 4: Therefore, the final result is that the sum of these terms equals three times the area of the triangle.