In triangle `A B C ,` if `cotA ,cotB ,cotC` are in `AdotPdot,` then `a^2,b^2,c^2` are in ____________ progression.

Answer

To solve the problem, we need to show that if \( \cot A, \cot B, \cot C \) are in arithmetic progression (AP), then \( a^2, b^2, c^2 \) are also in AP. Here’s a step-by-step solution:

### Step 1: Understand the Given Condition
We know that \( \cot A, \cot B, \cot C \) are in AP. This means:
\[ 2 \cot B = \cot A + \cot C \]

### Step 2: Express Cotangent in Terms of Sine and Cosine
Recall that \( \cot A = \frac{\cos A}{\sin A} \), \( \cot B = \frac{\cos B}{\sin B} \), and \( \cot C = \frac{\cos C}{\sin C} \). Thus, we can rewrite the condition:
\[ 2 \frac{\cos B}{\sin B} = \frac{\cos A}{\sin A} + \frac{\cos C}{\sin C} \]

### Step 3: Use the Sine Rule
According to the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \quad \text{(some constant)} \]
From this, we can express \( \sin A, \sin B, \sin C \) in terms of \( a, b, c \):
\[ \sin A = \frac{a}{k}, \quad \sin B = \frac{b}{k}, \quad \sin C = \frac{c}{k} \]

### Step 4: Substitute Sine Values into Cotangent Expressions
Substituting these into the cotangent expressions gives:
\[ \cot A = \frac{\cos A}{\frac{a}{k}} = \frac{k \cos A}{a}, \quad \cot B = \frac{k \cos B}{b}, \quad \cot C = \frac{k \cos C}{c} \]

### Step 5: Substitute into the AP Condition
Substituting these into the AP condition:
\[ 2 \frac{k \cos B}{b} = \frac{k \cos A}{a} + \frac{k \cos C}{c} \]
We can cancel \( k \) (assuming \( k \neq 0 \)):
\[ 2 \frac{\cos B}{b} = \frac{\cos A}{a} + \frac{\cos C}{c} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 2 \cos B \cdot ac = \cos A \cdot bc + \cos C \cdot ab \]

### Step 7: Using the Cosine Rule
Using the cosine rule, we know:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting these into the equation will lead to a relationship involving \( a^2, b^2, c^2 \).

### Step 8: Show that \( a^2, b^2, c^2 \) are in AP
After substituting and simplifying, we can show that:
\[ b^2 + c^2 - a^2, \quad a^2 + c^2 - b^2, \quad a^2 + b^2 - c^2 \]
are in AP, which implies that \( a^2, b^2, c^2 \) are also in AP.

### Conclusion
Thus, we conclude that if \( \cot A, \cot B, \cot C \) are in arithmetic progression, then \( a^2, b^2, c^2 \) are also in arithmetic progression.

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