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If I select Rotate tool, I can select center of rotation then and move it. After that I am able to rotate object around that center. But this rotation is imprecise. Suppose I want to rotate for 120 degrees. I can double click on Rotate tool and get Rotate dialog, where I can enter precise value in degrees. But this way I can't use shifted center of rotation because it jumps back to geometric center at the moment I call Rotate dialog.

Is it possible to rotate by precise angle with dialog and have shifted center of rotation.

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With the Rotate tool active, hold down the Alt key ( on OS X) then click the point you want to rotate the object around. The Rotate dialog box will then pop up and you can enter your precise angle.

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Will my rotation point snap precisely to intersection of guidelines in this situation? – Suzan Cioc Dec 22 '12 at 17:36
If you have Smart Guides on, then yes. With that enabled, you should see "intersect" pop up when you hover over the intersection of the guides: – JohnB Dec 22 '12 at 17:44

If you hit ctrl D after the first rotation copy, you can duplicate the object around the center. Useful for diving a circle into slices.

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