Question

In $ \triangle ABC $, if $ r $ is the inradius and $ r_1, r_2, r_3 $ are the ex-radii, then \[ \frac{1}{4} \left[ b^2 \sin 2C + c^2 \sin 2B \right] = \]

• A. \( rr_1 \tan \frac{A}{2} \)
• B. \( bc \cos A \)
• C. \( r_1 r_2 r_3 \)
• D. \( rr_1 \cot \frac{A}{2} \)

Hint:

In problems involving the inradius and ex-radii, utilize the relationship between the sides and angles of the triangle and apply trigonometric identities to simplify the expression.

Answer 1

The Correct Option is D

We are given the inradius \( r \) and the ex-radii \( r_1, r_2, r_3 \). We need to evaluate the expression \( \frac{1}{4} \left[ b^2 \sin 2C + c^2 \sin 2B \right] \).
Step 1: The general identity for the area of a triangle in terms of its sides and angles is: \[ \Delta = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B \] This identity allows us to relate the sides \( b \), \( c \), and angles \( A \), \( B \), and \( C \).
Step 2: The given expression involves the angles and sides of the triangle. We can express these terms using the trigonometric identities and ex-radii. Simplifying the given expression \( \frac{1}{4} \left[ b^2 \sin 2C + c^2 \sin 2B \right] \) results in: \[ \frac{1}{4} \left[ b^2 \sin 2C + c^2 \sin 2B \right] = rr_1 \cot \frac{A}{2} \] Step 3: Therefore, the simplified expression is \( rr_1 \cot \frac{A}{2} \).