In `Delta ABC`, if `r_(1) lt r_(2) lt r_(3)`, then find the order of lengths of the sides

Explanation

To solve the problem, we need to find the order of the lengths of the sides of triangle ABC given the condition \( r_1 < r_2 < r_3 \). Here’s the step-by-step solution:

### Step 1: Understand the relationship between inradii and sides
In a triangle, the inradii \( r_1, r_2, r_3 \) are given by the formulas:
- \( r_1 = \frac{\Delta}{s - a} \)
- \( r_2 = \frac{\Delta}{s - b} \)
- \( r_3 = \frac{\Delta}{s - c} \)

where \( \Delta \) is the area of the triangle, and \( s \) is the semi-perimeter defined as \( s = \frac{a + b + c}{2} \).

### Step 2: Set up the inequalities
Given the condition \( r_1 < r_2 < r_3 \), we can write:
\[ \frac{\Delta}{s - a} < \frac{\Delta}{s - b} < \frac{\Delta}{s - c} \]

### Step 3: Invert the inequalities
Since \( \Delta > 0 \) (the area of the triangle is positive), we can safely invert the inequalities:
\[ \frac{1}{s - a} > \frac{1}{s - b} > \frac{1}{s - c} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ s - b > s - a > s - c \]

### Step 5: Simplify the inequalities
By simplifying the inequalities, we get:
\[ -a > -b > -c \]

### Step 6: Multiply by -1
Multiplying the entire inequality by -1 (and reversing the order of the inequalities) gives:
\[ b > a > c \]

### Conclusion
Thus, the order of the lengths of the sides of triangle ABC is:
\[ c < a < b \]