In a `triangleABC`, if `3tan (A/2) tan (C /2)=1`, then sides a,b,c are in

Options

A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer

To solve the problem, we need to analyze the given equation involving the tangents of half-angles in triangle \( ABC \) and relate it to the sides of the triangle.

### Step-by-Step Solution:

1. **Given Equation**: We start with the equation:
\[ 3 \tan\left(\frac{A}{2}\right) \tan\left(\frac{C}{2}\right) = 1 \]

2. **Using the Half-Angle Tangent Formula**: The tangent of half-angles can be expressed in terms of the area of the triangle and its semi-perimeter:
\[ \tan\left(\frac{A}{2}\right) = \frac{\Delta}{s(s-a)} \quad \text{and} \quad \tan\left(\frac{C}{2}\right) = \frac{\Delta}{s(s-c)} \]
where \( \Delta \) is the area of triangle \( ABC \) and \( s \) is the semi-perimeter given by \( s = \frac{a+b+c}{2} \).

3. **Substituting into the Equation**: Substitute these expressions into the original equation:
\[ 3 \left(\frac{\Delta}{s(s-a)}\right) \left(\frac{\Delta}{s(s-c)}\right) = 1 \]
This simplifies to:
\[ \frac{3\Delta^2}{s^2(s-a)(s-c)} = 1 \]

4. **Rearranging the Equation**: Rearranging gives:
\[ 3\Delta^2 = s^2(s-a)(s-c) \]

5. **Using the Area Formula**: We know that the area \( \Delta \) can also be expressed as:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Therefore, substituting this into our equation gives:
\[ 3(s(s-a)(s-b)(s-c)) = s^2(s-a)(s-c) \]

6. **Dividing Both Sides**: We can divide both sides by \( (s-a)(s-c) \) (assuming \( s-a \neq 0 \) and \( s-c \neq 0 \)):
\[ 3s(s-b) = s^2 \]

7. **Rearranging Further**: This simplifies to:
\[ 3s - s = 3b \quad \Rightarrow \quad 2s = 3b \]

8. **Substituting for \( s \)**: Since \( s = \frac{a+b+c}{2} \), we substitute:
\[ 2 \left(\frac{a+b+c}{2}\right) = 3b \quad \Rightarrow \quad a + b + c = 3b \]

9. **Final Rearrangement**: Rearranging gives:
\[ a + c = 2b \]

10. **Conclusion**: The condition \( a + c = 2b \) indicates that the sides \( a, b, c \) are in Arithmetic Progression (AP).

### Final Answer:
The sides \( a, b, c \) are in **AP** (Arithmetic Progression).

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