Euclid's Elements Book I

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

Proposition I.22 · Problem

Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let the three given straight lines be A, B, C, of which any two taken together are greater than the remaining one. It is required to construct a triangle out of three straight lines equal to A, B, C. (Drag the right ends of the lines to change their lengths.)
  2. Let there be set out a straight line DE, limited at D but unlimited in the direction of E.
  3. On DE let DF be made equal to A, FG equal to B, and GH equal to C.
    Proposition I.3
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  4. With centre F and distance FD let the circle DKL be described.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  5. With centre G and distance GH let the circle KLH be described.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  6. The circles meet at K; let KF, KG be joined. The claim: the triangle KFG has been constructed out of three straight lines equal to A, B, C.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  7. Since the point F is the centre of the circle DKL, FD is equal to FK. But FD is equal to A; therefore KF is also equal to A.
    Definition 15Common Notion 1
    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.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
  8. Again, since the point G is the centre of the circle LKH, GH is equal to GK. But GH is equal to C; therefore KG is also equal to C.
    Definition 15Common Notion 1
    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.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
  9. And FG is also equal to B. Therefore the three straight lines KF, FG, GK are equal to A, B, C, and the triangle KFG has been constructed.