To solve this problem, we’ll use the properties of right triangles and the relationship between the altitude and the sides. Here's the approach:

1. Apply the Altitude Property in a Right Triangle: For a right triangle $ABC$ with a right angle at $C$, if the altitude from $C$ intersects $AB$ at $D$, then: $CD = \frac{a \cdot b}{c}$ where $a$, $b$, and $c$ are the side lengths of the triangle opposite vertices $A$, $B$, and the hypotenuse $AB$ respectively.

2. Use Given Information: We know $CD = 12$ and that $a$, $b$, and $c$ are integers, which implies that: $\frac{a \cdot b}{c} = 12$ Therefore, $a \cdot b = 12c$

3. Minimum Integer Perimeter: We aim to find integer values of $a$, $b$, and $c$ such that $a^2 + b^2 = c^2$, $a \cdot b = 12c$, and the perimeter $P = a + b + c$ is minimized.

4. Testing Small Integer Values: By testing possible values, we find that one solution is:

   • $a = 15$
   • $b = 20$
   • $c = 25$

   Here: $a \cdot b = 15 \cdot 20 = 300 = 12 \cdot 25$ and $a^2 + b^2 = 15^2 + 20^2 = 225 + 400 = 625 = 25^2 = c^2$

5. Calculate the Perimeter: The perimeter $P = a + b + c = 15 + 20 + 25 = 60$.

Thus, the minimum possible perimeter of triangle $ABC$ is: $\boxed{60}$

Would you like a detailed breakdown of each calculation or additional help?

Here are some related questions for further exploration:

1. How does the altitude property in a right triangle help in determining integer side lengths?
2. What other integer side lengths satisfy the condition $\frac{a \cdot b}{c} = 12$?
3. How can we generalize the relationship for different altitudes in right triangles with integer sides?
4. What other properties of right triangles involve integer side lengths and altitudes?
5. How would the problem change if $CD$ was a different integer?

Tip: Right triangles with integer side lengths are known as Pythagorean triples, often helpful in geometry problems involving integer constraints.