To solve the problem, let's analyze the figure and the given information:

Problem Breakdown:

1. Figure Description:

   • $ADEF$ is a rectangle.
   • $OABC$ is a square.
   • Lines $FD$ and $AE$ intersect at $O$.
   • $AF = x$, $AE = y$, and $ED = z$.

2. Given Conditions:

   • $FD$ and $AE$ intersect at $O$.
   • $AF = OF$ (since $OABC$ is a square, $O$ is the center, and diagonals of a square bisect each other).

Steps to Solve:

1. Coordinates and Relationships:

   • Let the square $OABC$ have side length $a$.
   • $AF = x$, $AE = y$, and $ED = z$.

   Since $OABC$ is a square and $O$ is the center, the coordinates of $O$ are the midpoint of diagonals $AC$ and $BD$.

   Because $ADEF$ is a rectangle and $OABC$ is a square, the diagonal properties will help us relate $x$, $y$, and $z$ using the Pythagorean theorem.

2. Apply Geometric Relationships:

   • Since $OABC$ is a square, $OA = OB = OC = OD = \frac{\sqrt{2}}{2}a$.
   • $AF = x$ and $AE = y$ implies that triangle $OAF$ is isosceles, so: $AF = FO = x$ And since $ADEF$ is a rectangle: $AE = ED = y = z$ This implies: $x^2 + y^2 = z^2$
   • Given $FD$ intersects $AE$ at $O$, we have an additional relationship $x + z = 2a$.

Solution:

Now, solving the system of equations: $x^2 + y^2 = z^2$ and $x + z = 2a$

The specific values of $x$, $y$, and $z$ would depend on solving these equations given the specific measures or deriving them from the geometric properties of the square and rectangle.

Let me know if you'd like a more detailed calculation or explanation on a specific part. Would you like the exact coordinates and detailed steps, or any clarification?

Relative Questions:

1. How would the answer change if the square were not centered on the rectangle?
2. What is the effect of increasing the size of the square on the lengths $x$, $y$, and $z$?
3. How do properties of rectangles and squares influence the relationships between $x$, $y$, and $z$?
4. Can you derive another geometric figure with similar properties using different starting shapes?
5. What would happen if $OABC$ were not a square but a different quadrilateral?

Tip:

When solving geometric problems involving squares and rectangles, remember to use symmetry and the properties of diagonals effectively to simplify the problem.