Euclid's Proposition 2: Given a line segment and an end point, a segment of the same length can be constructed.

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Construction

  1. Let A be the point and BC be the line.
  2. Construct a line segment on AB. Postulate 1
  3. Construct an equilateral triangle on AB. Mark the third vertex of the triangle D. Proposition 1
  4. Extend line segments DA and DB. Postulate 2
  5. Now draw a circle with B as the center and BC as the radius. Mark the point were the circle intersects the extension of DB as G. Postulate 3
  6. Now draw a circle with D as center and radius DG. Postulate 3
  7. Since B is the center of the circle containing C and G, BC is congruent with BG. Also, for the same reason, DE is congruent with DF.
  8. But since DA is congruent with DB, and DF is congruent with DE, then AF is congruent with BC.

Notes:

Note that if we change the length of BC, the length of AF changes too. However, if we change the position of B, or the position of A on the circle without changing the size of the circle, the length of AF remains constant. This is because the length of AF depends only on the length of BC, not the location or orientation of BC.

Note also that we can change the position of AF by moving A without changing the length. Also, we can change the orientation of AF by moving F around the circle without changing the length of AF. This is because a segment can be copied to any location or having any orientation without changing the length.

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Euclid's Elements

http://babbage.clarku.edu/~djoyce/java/elements/toc.html