Ruler and Compass Construction

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Starting Conditions

  1. A plane with two distinct points, one labelled 0 and the other labelled 1.
  2. A marked ruler — just a straight-edge.
  3. A compass that collapses when lifted — can’t be used to copy a distance.

Question

What points, shapes, and figures can be constructed using only the above?

Possible Operations

  1. Use the ruler to draw the line through two points.
  2. Use the compass to draw the circle with centre an already constructed point and going through another point.

Creating New Points

New points are created at:

  1. An intersection of two lines.
  2. An intersection of a line and a circle.
  3. An intersection of two circles.

The Point at 2

  1. Draw the line through 0 and 1.
  2. Draw the circle through 0 with centre 1.
  3. Then the other point where the line and circle intersect is twice as far from 0 as 1 and is collinear with 0 and 1.