Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.2 · Problem

To place at a given point a straight line equal to a given straight line.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let A be the given point, and BC the given straight line. It is required to place at the point A a straight line equal to BC.
  2. From the point A to the point B let the straight line AB be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  3. On AB let the equilateral triangle DAB be constructed.
    Proposition I.1
    Proposition I.1: On a given finite straight line to construct an equilateral triangle.
  4. Let the straight lines AE, BF be produced in a straight line with DA, DB.
    Postulate 2
    Postulate 2: To produce a finite straight line continuously in a straight line.
  5. With centre B and distance BC let the circle CGH be described; it meets BF at G.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  6. Again, with centre D and distance DG let the circle GKL be described; it meets AE at L.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  7. Since the point B is the centre of the circle CGH, BC is equal to BG.
    Definition 15
    Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
  8. Again, since the point D is the centre of the circle GKL, DL is equal to DG.
    Definition 15
    Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
  9. And in these DA is equal to DB; subtracting DA, DB from the equals DL, DG, the remainder AL is equal to the remainder BG.
    Common Notion 3
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.
  10. But BC was also proved equal to BG; therefore each of the straight lines AL, BC is equal to BG. Things which are equal to the same thing are also equal to one another; therefore AL is also equal to BC. So at the given point A the straight line AL has been placed equal to the given straight line BC.
    Common Notion 1
    Common Notion 1: Things which are equal to the same thing are also equal to one another.