Given:
We are given that a, b, c are integers such that |a-b|^19 * |c-a|^19 = 1.

To Find:
We need to find the value of |c-a| * |a-b| * |b-c|.

Approach:
Let's break down the given equation and identify some key points:

Key Point 1:
The only way the product of two integers can be equal to 1 is if both the integers are either 1 or -1.

Key Point 2:
According to the given equation, |a-b|^19 * |c-a|^19 = 1. This implies that both |a-b|^19 and |c-a|^19 must be either 1 or -1.

Key Point 3:
Since |a-b|^19 = 1 or -1, it means that |a-b| can be either 1 or -1. Similarly, |c-a| can also be either 1 or -1.

Key Point 4:
Now, let's consider the third factor, |b-c|. We need to find its value.

Case 1: |a-b| = 1 and |c-a| = 1
If both |a-b| and |c-a| are equal to 1, then |b-c| must also be equal to 1. In this case, |c-a| * |a-b| * |b-c| = 1 * 1 * 1 = 1.

Case 2: |a-b| = -1 and |c-a| = -1
If both |a-b| and |c-a| are equal to -1, then |b-c| must also be equal to 1. In this case, |c-a| * |a-b| * |b-c| = -1 * -1 * 1 = 1.

Case 3: |a-b| = 1 and |c-a| = -1
If |a-b| is equal to 1 and |c-a| is equal to -1, then |b-c| must be equal to -1. In this case, |c-a| * |a-b| * |b-c| = -1 * 1 * -1 = 1.

Case 4: |a-b| = -1 and |c-a| = 1
If |a-b| is equal to -1 and |c-a| is equal to 1, then |b-c| must be equal to -1. In this case, |c-a| * |a-b| * |b-c| = 1 * -1 * -1 = 1.

Conclusion:
In all four cases, the value of |c-a| * |a-b| * |b-c| is equal to 1.