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Let's analyze each statement given the properties visible in the straightedge and compass construction of the diagram:

A. **AC = BC**: Since B is the center of the circle in which point C lies, BC is a radius. Similarly, A being the center of the circle in which point C lies makes AC a radius. As both AC and BC are radii of their respective circles, **AC = BC is true**.

B. **AC = BD**: This statement can be evaluated based on symmetry and the properties of the circles. Since both AC and BD represent the radius of the same circle centered at B, **AC = BD is also true**.

C. **CD = AB**: There isn't a straightforward symmetry or equal segment argument that can be derived just from the construction description, so without additional information on the specific lengths or constructions, we would consider this uncertain. However, considering typical constructions, this isn't necessarily true.

D. **ABCD is a square**: For ABCD to be a square, all sides would need to be equal, and all angles would need to be right angles. We can't definitively conclude from the given diagram that all angles are right angles; thus, this statement is likely **false**.

E. **ABD is an equilateral triangle**: For triangle ABD to be equilateral, AB, BD, and AD would need to be of equal length. As AD is not given as a radius or specified as equal to AB or BD in the standard construction rules, this is **unlikely to be true**.

F. **CD = AB + AB**: This suggests CD should be twice the length of AB. Given CD and AB are separate segments without clear proportional indications or relationships via the radii, this statement also appears **false**.

The true statements based on the given construction and standard geometrical principles are:
- **A: AC = BC**
- **B: AC = BD**

Would you like to know more details about any specific part of this construction, or have any other questions on this topic? Here are some relative questions:
1. How is it determined that AC = BC and AC = BD from the diagram?
2. What additional construction could make CD = AB true?
3. Why can’t ABCD be a square based on the information provided?
4. How to prove a triangle is equilateral using geometric properties?
5. What is the significance of a straightedge and compass construction in geometry?

Tip: Remember, when analyzing geometry problems, properties such as equal segment lengths, symmetry, and angle measures based on circles (radii, diameter) often provide crucial clues.

This diagram is a straightedge and compass construction. A is the center of one circle, and B is the center of the other. Select all the true statements.

This problem involves analyzing the true statements from a compass and straightedge construction, with concepts such as equal radii, symmetry, and geometric shapes. Suitable for high school students, it explores the properties of circles and geometric constructions.

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