To solve the problem, we start with the given equation:
\[
\frac{\tan(A/2)}{\tan(C/2)} = \frac{1}{2}
\]
### Step 1: Use the half-angle tangent formulas
We know the formulas for \(\tan(A/2)\) and \(\tan(C/2)\):
\[
\tan\left(\frac{A}{2}\right) = \sqrt{\frac{s - b}{s(s - a)}}
\]
\[
\tan\left(\frac{C}{2}\right) = \sqrt{\frac{s - a}{s(s - c)}}
\]
where \(s = \frac{a + b + c}{2}\) is the semi-perimeter of triangle \(ABC\).
### Step 2: Substitute the formulas into the equation
Substituting these formulas into the given equation, we have:
\[
\frac{\sqrt{\frac{s - b}{s(s - a)}}}{\sqrt{\frac{s - a}{s(s - c)}}} = \frac{1}{2}
\]
### Step 3: Simplify the equation
This simplifies to:
\[
\frac{\sqrt{(s - b)(s - c)}}{\sqrt{(s - a)(s - a)}} = \frac{1}{2}
\]
Squaring both sides gives:
\[
\frac{(s - b)(s - c)}{(s - a)(s - a)} = \frac{1}{4}
\]
### Step 4: Cross-multiply
Cross-multiplying leads to:
\[
4(s - b)(s - c) = (s - a)^2
\]
### Step 5: Expand both sides
Expanding both sides, we have:
\[
4(s^2 - (b+c)s + bc) = s^2 - 2as + a^2
\]
### Step 6: Rearranging the equation
Rearranging gives:
\[
4s^2 - 4(b+c)s + 4bc = s^2 - 2as + a^2
\]
Combining like terms results in:
\[
3s^2 - (4b + 4c - 2a)s + (4bc - a^2) = 0
\]
### Step 7: Analyze the quadratic equation
For \(a\), \(b\), and \(c\) to be in an arithmetic progression (AP), the discriminant of this quadratic must be zero.
### Step 8: Check for conditions of AP, GP, HP
To check if \(a\), \(b\), and \(c\) are in AP, GP, or HP, we can analyze the relationships derived from the quadratic equation.
After simplification, we find:
\[
A + C - 4B = 0
\]
This does not satisfy the conditions for AP, GP, or HP.
### Conclusion
Thus, the values of \(a\), \(b\), and \(c\) are not in AP, GP, or HP. Therefore, the answer is:
**None of these.**
---
