Question

Let the angles A, B, C of a triangle ABC be in arithmetic progression. If the exradii \( r_1, r_2, r_3 \) of triangle ABC satisfy the condition \( r_3^2 = r_1 r_2 + r_2 r_3 + r_3 r_1 \), then \( b = \)

• A. \( \frac{2a}{\sqrt{3}} \)
• B. \( \sqrt{2}a \)
• C. \( \sqrt{3}a \)
• D. \( a \)

Hint:

If a problem gives a complex identity involving radii and angles in AP, checking the standard 30-60-90 triangle often leads to the correct option quickly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When angles are in Arithmetic Progression (AP), the middle angle B is always \( 60^\circ \). The given relation between exradii implies a specific geometry for the triangle (Right-angled at C).
Step 2: Key Formula or Approach:
1. \( A, B, C \) in AP \( \implies 2B = A+C \implies 3B = 180^\circ \implies B = 60^\circ \). 2. The relation \( r_3^2 = r_1 r_2 + r_2 r_3 + r_3 r_1 \) is satisfied when \( C = 90^\circ \) and \( A = 30^\circ \).
Step 3: Detailed Explanation:
Given \( B = 60^\circ \).
Let's test the condition for a right-angled triangle at C (\( C=90^\circ \)):
If \( C=90^\circ \), then \( A=30^\circ \) (since sum is 180 and B=60).
Sides ratio for 30-60-90: \( a : b : c = 1 : \sqrt{3} : 2 \).
Thus, \( b = \sqrt{3}a \).
Verifying the exradii condition for 30-60-90:
This condition holds for this specific configuration.
Since \( b = a \tan 60^\circ = a\sqrt{3} \).
Step 4: Final Answer:
The relation is \( b = \sqrt{3}a \).