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gear How can I quickly select those two edges all the way around the circle?

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First suggestion: work under an Array modifier, based on a rotated Empty at the wheel's center. That way, all adjustments you make to one cog will be reflected in all the others.

But you could make a reasonably quick selection:

  • Select one of the edges of interest
  • ShiftG > select similar > direction
  • In an orthographic view, using circle-select, enclose the unwanted (inner) edges in that direction, and middle-click, deselect those.
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You can use Checker Deselect for this.

  1. Select all the edges around using Select > Edge Rings
  2. Deselect one of the edges (Shift + LMB) to break the loop
  3. Use Select > Checker Deselect (play with the values, and adjust offset if needed)
  4. Add that one edge to selection (again using Shift + LMB)

enter image description here

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