Question

In \( \triangle ABC \), if \( r_1 = 2r_2 = 3r_3 \), then

• A. \( b + c = 2a \)
• B. \( a + b = 2c \)
• C. \( a + c = 2b \)
• D. \( A = abc \)

Hint:

Use the formulas for the exradii \( r_1 = \frac{\Delta}{s - a}, r_2 = \frac{\Delta}{s - b}, r_3 = \frac{\Delta}{s - c} \). Set up equations based on the given relation \( r_1 = 2r_2 = 3r_3 \). Solve for \( s - a, s - b, s - c \) in terms of a common variable. Then find \( a, b, c \) in terms of the semi-perimeter \( s \) and check which of the given options holds true.

Answer 1

The Correct Option is C

We know the formulas for the exradii \( r_1, r_2, r_3 \) of a triangle \( ABC \) with semi-perimeter \( s \) and area \( \Delta \): $$ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c} $$ Given \( r_1 = 2r_2 = 3r_3 \).
Let \( r_1 = 6k \) for some constant \( k>0 \).
Then \( r_2 = 3k \) and \( r_3 = 2k \).
$$ \frac{\Delta}{s - a} = 6k \implies s - a = \frac{\Delta}{6k} $$ $$ \frac{\Delta}{s - b} = 3k \implies s - b = \frac{\Delta}{3k} $$ $$ \frac{\Delta}{s - c} = 2k \implies s - c = \frac{\Delta}{2k} $$ Let \( \frac{\Delta}{k} = x \).
Then \( s - a = \frac{x}{6}, s - b = \frac{x}{3}, s - c = \frac{x}{2} \).
Adding these three equations: $$ (s - a) + (s - b) + (s - c) = \frac{x}{6} + \frac{x}{3} + \frac{x}{2} $$ $$ 3s - (a + b + c) = \frac{x + 2x + 3x}{6} = \frac{6x}{6} = x $$ Since \( a + b + c = 2s \), we have \( 3s - 2s = s = x \).
Now substitute \( s = x \) back into the expressions for \( s - a, s - b, s - c \): $$ s - a = \frac{s}{6} \implies a = s - \frac{s}{6} = \frac{5s}{6} $$ $$ s - b = \frac{s}{3} \implies b = s - \frac{s}{3} = \frac{2s}{3} = \frac{4s}{6} $$ $$ s - c = \frac{s}{2} \implies c = s - \frac{s}{2} = \frac{s}{2} = \frac{3s}{6} $$ So, the sides are in the ratio \( a : b : c = \frac{5s}{6} : \frac{4s}{6} : \frac{3s}{6} = 5 : 4 : 3 \).
Let \( a = 5m, b = 4m, c = 3m \) for some constant \( m>0 \).
Check the options:
Option (A) \( b + c = 4m + 3m = 7m \), \( 2a = 10m \).
\( 7m \neq 10m \).

Option (B) \( a + b = 5m + 4m = 9m \), \( 2c = 6m \).
\( 9m \neq 6m \).

Option (C) \( a + c = 5m + 3m = 8m \), \( 2b = 8m \).
\( 8m = 8m \).

Option (D) \( A = abc \) is not a valid relation between sides.