`r_1 cot ""A/2=r_2 cot""B/2=r_3 cot""C/2=S`

Answer

To solve the equation \( r_1 \cot \frac{A}{2} = r_2 \cot \frac{B}{2} = r_3 \cot \frac{C}{2} = S \), we will analyze each term separately and derive the relationships.

### Step 1: Understand the terms

We have:
- \( r_1 \) is the radius of the incircle (inradius).
- \( r_2 \) is the radius of the A-excircle.
- \( r_3 \) is the radius of the B-excircle.
- \( S \) is the area of the triangle.

### Step 2: Use the formula for \( r_1 \)

The formula for the inradius \( r_1 \) is given by:
\[ r_1 = \frac{S}{s} \]
where \( S \) is the area of the triangle and \( s \) is the semi-perimeter defined as:
\[ s = \frac{a + b + c}{2} \]

### Step 3: Use the formula for \( \cot \frac{A}{2} \)

The formula for \( \cot \frac{A}{2} \) is:
\[ \cot \frac{A}{2} = \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} \]

### Step 4: Substitute into the equation for \( r_1 \)

Substituting \( r_1 \) into the equation:
\[ r_1 \cot \frac{A}{2} = \frac{S}{s} \cdot \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} \]

### Step 5: Analyze \( r_2 \) and \( r_3 \)

Similarly, we can write:
\[ r_2 = \frac{S}{s-a}, \quad r_3 = \frac{S}{s-b} \]
and their respective cotangent terms:
\[ \cot \frac{B}{2} = \sqrt{\frac{s(s-b)}{(s-a)(s-c)}}, \quad \cot \frac{C}{2} = \sqrt{\frac{s(s-c)}{(s-a)(s-b)}} \]

### Step 6: Set up the equations

Now, we can set up the equations:
\[ r_2 \cot \frac{B}{2} = \frac{S}{s-a} \cdot \sqrt{\frac{s(s-b)}{(s-a)(s-c)}} \]
\[ r_3 \cot \frac{C}{2} = \frac{S}{s-b} \cdot \sqrt{\frac{s(s-c)}{(s-a)(s-b)}} \]

### Step 7: Equate all three expressions

Now we equate:
\[ \frac{S}{s} \cdot \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} = \frac{S}{s-a} \cdot \sqrt{\frac{s(s-b)}{(s-a)(s-c)}} = \frac{S}{s-b} \cdot \sqrt{\frac{s(s-c)}{(s-a)(s-b)}} \]

### Step 8: Analyze the conditions

From these equalities, we can analyze the conditions under which they hold true. We can conclude that:
- Each expression must be equal to \( S \) under the conditions of the triangle.

### Conclusion

Thus, we conclude that \( r_1 \cot \frac{A}{2} = r_2 \cot \frac{B}{2} = r_3 \cot \frac{C}{2} \) does not equal \( S \) under typical triangle conditions, leading us to the conclusion that the statement is false.