If, in a `triangleABC, tan(A/2) =5/6` and `tan(C/2) = 2/5`, then:

Options

A. a,b,c are in A.P.
B. a,c,b are in A.P.
C. b,a,c are in A.P.
D. a,b,c are in G.P.

Answer

To solve the problem, we need to determine the relationship between the angles \(A\), \(B\), and \(C\) in triangle \(ABC\) given that \(\tan\left(\frac{A}{2}\right) = \frac{5}{6}\) and \(\tan\left(\frac{C}{2}\right) = \frac{2}{5}\).

### Step-by-Step Solution:

1. **Use the given values**:
We have:
\[ \tan\left(\frac{A}{2}\right) = \frac{5}{6} \]
\[ \tan\left(\frac{C}{2}\right) = \frac{2}{5} \]

2. **Calculate the product of the tangents**:
We multiply the two tangent values:
\[ \tan\left(\frac{A}{2}\right) \cdot \tan\left(\frac{C}{2}\right) = \frac{5}{6} \cdot \frac{2}{5} = \frac{1}{3} \]

3. **Use the relationship from Heron's formula**:
According to Heron's formula, we can express \(\tan\left(\frac{A}{2}\right)\) and \(\tan\left(\frac{C}{2}\right)\) in terms of the semi-perimeter \(s\) and the sides \(a\), \(b\), and \(c\):
\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \]
\[ \tan\left(\frac{C}{2}\right) = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}} \]

4. **Set up the equation**:
From the product of the tangents:
\[ \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \cdot \sqrt{\frac{(s-a)(s-b)}{s(s-c)}} = \frac{1}{3} \]
Simplifying this gives:
\[ \frac{(s-b)(s-c)(s-a)(s-b)}{s^2(s-a)(s-c)} = \frac{1}{3} \]

5. **Simplify the equation**:
This simplifies to:
\[ (s-b)^2 = \frac{s^2}{3} \]
This implies:
\[ s-b = \frac{s}{\sqrt{3}} \]

6. **Express \(s\) in terms of \(b\)**:
Rearranging gives:
\[ s = b\left(1 + \frac{1}{\sqrt{3}}\right) \]

7. **Find the relationship between \(a\), \(b\), and \(c\)**:
The semi-perimeter \(s\) is given by:
\[ s = \frac{a + b + c}{2} \]
Therefore:
\[ a + b + c = 2s = 2b\left(1 + \frac{1}{\sqrt{3}}\right) \]

8. **Determine the relationship**:
From the derived expression, we can conclude:
\[ a = 2b - b\left(1 + \frac{1}{\sqrt{3}}\right) = b\left(1 - \frac{1}{\sqrt{3}}\right) \]
This indicates that \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP).

### Conclusion:
Thus, we conclude that the angles \(A\), \(B\), and \(C\) are in Arithmetic Progression (AP).