Euclid's Elements Book I

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

Proposition I.12 · Problem

To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let AB be the given infinite straight line, and C the given point which is not on it. It is required to draw a perpendicular from the point C to AB.
  2. Let a point D be taken at random on the other side of AB (you can drag it).
  3. With centre C and distance CD let the circle EFG be described; it cuts AB at E and G.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  4. Let EG be bisected at H.
    Proposition I.10
    Proposition I.10: To bisect a given finite straight line.
  5. Let CG, CH, CE be joined. The claim: CH is the perpendicular drawn from C to AB.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  6. Since GH is equal to HE, and HC is common, the two sides GH, HC are equal to the two sides EH, HC, and the base CG is equal to the base CE; therefore the angle CHG is equal to the angle EHC.
    Proposition I.8
    Proposition I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.
  7. And they are adjacent. When a straight line set up on a straight line makes the adjacent angles equal, each is right, and the line standing on the other is called a perpendicular. Therefore CH is the perpendicular drawn from C to AB.
    Definition 10
    Definition 10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.