How do I rotate multiple identical circles (any shape) around a single point?

Rotation Anchor

You can place the object's Rotation Anchor anywhere in the drawing, and drag any of the Rotation Handles to rotate around that Anchor.

To reach the Anchor and Handles, click the object twice (not double click).

Constrain Angle

While rotating (dragging the Rotation Handles), you can constrain the rotation Angle by pressing the Ctrl key.

To edit the Constrain Angle in X3, go to Tools > Options > Workspace > Edit.

Note
The Constrain Angle is a World angle (as apposed to Local); it's not relative to the object's current angle.

EDIT: Actually it is local; it is relative to the object's current angle.

Dynamic Guides

Dynamic Guides will efficiently help you in such jobs. To activate it, select nothing (click on empty space on the drawing), and click the Dynamic Guides button on the Property Bar.

To edit the Dynamic Guides' preferences, go to Tools > Options > Workspace > Dynamic Guides.

Rotate a Duplicate

While you rotate, you can click the right mouse button (a Plus sign appears) to apply the rotation to a duplicate and leave the original object in its place.

After you place the duplicate on a certain angle, you can press Ctrl + D to place another duplicate on the same angle increment.

Using the Constrain Angle, Dynamic Guides and Duplicates, you can easily place any number of identical objects along a circle;

1. Calculate the Angle (360/number of objects)
2. Edit the Constrain Angle (if needed)
3. Place the Anchor
4. Rotate a duplicate
5. Ctrl+Ds

Transformation Docker

Explore the great Transformation Docker (Arrange > Transformations > (Rotate))

This is a brief introduction; it's your turn to explore more options and details.