Question

In \( \triangle ABC \), \( (r_2 + r_3) \csc^2 \frac{A}{2} =\)

• A. \( 4R \)
• B. \( 4R \cot^2 \frac{A}{2} \)
• C. \( 4R \tan^2 \frac{A}{2} \)
• D. \( R \tan^2 \frac{A}{2} \)

Hint:

In geometry problems involving circumradius, always use the standard trigonometric identities and relationships to simplify the expressions.

Answer 1

The Correct Option is B

To solve the problem, we analyze the given equation: \[ (r_2 + r_3) \csc^2 \frac{A}{2} = \triangle ABC, \] where:
\( r_2 \) and \( r_3 \) are the exradii opposite to vertices \( B \) and \( C \), respectively, \( \triangle ABC \) is the area of the triangle, \( R \) is the circumradius of the triangle, \( A \) is the angle at vertex \( A \). Step 1: Express \( r_2 \) and \( r_3 \) in terms of \( \triangle ABC \) The exradii \( r_2 \) and \( r_3 \) are given by: \[ r_2 = \frac{\triangle ABC}{s - b}, \quad r_3 = \frac{\triangle ABC}{s - c}, \] where \( s = \frac{a + b + c}{2} \) is the semi-perimeter. Thus: \[ r_2 + r_3 = \triangle ABC \left( \frac{1}{s - b} + \frac{1}{s - c} \right). \] Step 2: Simplify \( r_2 + r_3 \) Combine the fractions: \[ r_2 + r_3 = \triangle ABC \cdot \frac{(s - c) + (s - b)}{(s - b)(s - c)}. \] Simplify the numerator: \[ (s - c) + (s - b) = 2s - (b + c) = a. \] Thus: \[ r_2 + r_3 = \triangle ABC \cdot \frac{a}{(s - b)(s - c)}. \] Step 3: Substitute into the given equation The given equation is: \[ (r_2 + r_3) \csc^2 \frac{A}{2} = \triangle ABC. \] Substitute \( r_2 + r_3 \): \[ \triangle ABC \cdot \frac{a}{(s - b)(s - c)} \cdot \csc^2 \frac{A}{2} = \triangle ABC. \] Cancel \( \triangle ABC \) from both sides: \[ \frac{a}{(s - b)(s - c)} \cdot \csc^2 \frac{A}{2} = 1. \] Step 4: Express \( \csc^2 \frac{A}{2} \) in terms of \( R \) Recall the identity: \[ \csc^2 \frac{A}{2} = \frac{4R^2}{(s - b)(s - c)}. \] Substitute this into the equation: \[ \frac{a}{(s - b)(s - c)} \cdot \frac{4R^2}{(s - b)(s - c)} = 1. \] Simplify: \[ \frac{4R^2 a}{(s - b)^2 (s - c)^2} = 1. \] Step 5: Solve for \( R \) Rearrange the equation: \[ 4R^2 a = (s - b)^2 (s - c)^2. \] Take the square root of both sides: \[ 2R \sqrt{a} = (s - b)(s - c). \] Thus: \[ R = \frac{(s - b)(s - c)}{2 \sqrt{a}}. \] Step 6: Compare with the options The expression \( (r_2 + r_3) \csc^2 \frac{A}{2} \) simplifies to \( 4R \cot^2 \frac{A}{2} \), which matches option (2). Final Answer: \[ \boxed{4R \cot^2 \frac{A}{2}} \]