We are given that the chords $\overline{AB}$ and $\overline{CD}$ intersect at point $P$, and that $AP = 3 \cdot PB$. Our goal is to find the length of $\overline{AB}$.

Let $AP = x$ and $PB = y$. According to the problem, we are given that:

$AP = 3 \cdot PB \quad \text{or} \quad x = 3y.$

Thus, the length of $AB$ is the sum of $AP$ and $PB$, which is:

$AB = AP + PB = x + y.$

Substituting $x = 3y$, we get:

$AB = 3y + y = 4y.$

Therefore, the length of $AB$ is $4y$, where $y = PB$. This expression represents the length of chord $AB$, but since no additional information (such as the lengths of the other segments) is provided, the final answer is $AB = 4y$.

Let me know if you need more details or any further questions. Here are some related queries you might consider:

1. What is the length of chord $CD$ given other conditions?
2. How can we determine the value of $y$ if more information is provided?
3. What is the geometric significance of the ratio $AP:PB$?
4. How does the Power of a Point theorem relate to this problem?
5. What additional properties of circles are useful for solving chord-related problems?

Tip: In circle problems involving intersecting chords, the Power of a Point theorem is often useful to relate the lengths of the segments formed.