To determine the measure of angle $\angle AOB$ formed by the points of tangency and the centers of the circles, let's analyze the given geometric configuration step-by-step.

Step-by-Step Analysis:

1. Understand the Configuration:

   • We have two circles that are tangent to each other.
   • Each circle is tangent to two sides of a square.
   • The centers of the circles, $O$, lie on the line segment $AB$ which is the diagonal of the square.

2. Label the Diagram:

   • Let the side length of the square be $s$.
   • Let the radius of each circle be $r$.
   • Since the circles are identical and tangent to each other, each circle's diameter is equal to $s/2$, i.e., $2r = s/2 \implies r = s/4$.

3. Diagonal of the Square:

   • The length of the diagonal $AB$ of the square is given by $AB = s\sqrt{2}$.

4. Position of Points:

   • Points $A$ and $B$ are points of tangency where the circles touch the sides of the square.
   • Points $O$ are the centers of the circles.

5. Triangle $AOB$:

   • The line $AB$ passes through the centers of the two circles, making it the diagonal of the square.
   • The circles being tangent to each other means that the distance between their centers (point $O$) is equal to $2r$.

6. Calculating the Angle $\angle AOB$:

   • Notice that $\angle AOB$ is the angle between the two radii of the circles that meet at the point of tangency between the two circles.
   • Since the circles are symmetric about the diagonal of the square, this configuration forms a right triangle.

Therefore, $\angle AOB$ is the angle subtended by the radii at the point where they touch. Given the symmetry and tangency points on the square's diagonal, $\angle AOB$ will be 45°.

Conclusion: The measure of the angle $\angle AOB$ is $45^\circ$.

Would you like further details or have any questions?

Here are 8 related questions:

1. How do you find the radius of circles tangent to a square?
2. How do you prove that two circles are tangent to each other?
3. How do you calculate the diagonal of a square?
4. How does symmetry in geometry help simplify problems?
5. What are some properties of tangent circles?
6. How do you use trigonometry to solve geometric problems?
7. What are the properties of angles formed by tangents and radii in circles?
8. How do you determine the measure of angles in complex geometric configurations?

Tip: When dealing with tangency and symmetry in geometry, breaking down the problem into simpler shapes like triangles and identifying symmetries can greatly simplify the solution.