The question appears to involve calculating specific points on a segment and proportional distances between points, given that:

1. Segment $AB$ is on the line containing the point $C(10, -10)$.
2. A point lies on this segment such that it is $\frac{7}{3}$ of the distance from one endpoint of the segment.

Additionally:

• It's given that the distance from point $A$ to point $B$ should satisfy certain conditions.
• The coordinates for points $A$ and $B$ must be calculated to form a reflection or symmetrical arrangement relative to point $C$.

Would you like a step-by-step solution, or do you have specific questions about certain parts of this problem?

Here are some related questions you might consider:

1. How do you calculate the coordinates of a point on a line segment at a specified distance ratio?
2. What formula should be used to determine the distance between two points?
3. How can symmetry relative to a given point be determined?
4. How does the direction of a vector affect proportional distance calculations?
5. How can we confirm that a calculated point lies on a specific line?

Tip: When working with line segments and distances, always double-check calculations for proportional distances to ensure accuracy.