Question

In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):

• A. \( a + b \)
• B. \( c + a \)
• C. \( 2 \sqrt{2} R \cos \left( \frac{C - A}{2} \right) \)
• D. \( 2 \sqrt{2} R \cos \left( \frac{B - C}{2} \right) \)

Hint:

For equations involving inradius and circumradius, use geometric identities and properties to simplify the given relations. Often, trigonometric functions help relate angles and side lengths.

Answer 1

The Correct Option is D

We are given the equation: \[ (r_2 - r_1)(r_3 - r_1) = 2r_2r_3. \] This is related to the semi-perimeter and other geometrical properties of the triangle. To solve for \( 2(r + R) \), we apply various geometric and trigonometric relationships, leading to the formula: \[ 2(r + R) = 2 \sqrt{2} R \cos \left( \frac{B - C}{2} \right). \] Thus, the correct answer is \( 2 \sqrt{2} R \cos \left( \frac{B - C}{2} \right) \).