Proposition I.12

To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.1

Let AB be the given infinite straight line, and C the given point which is not on it; thus it is required to draw to the given infinite straight line AB, from the given point C which is not on it, a perpendicular straight line.

For let a point D be taken at random on the other side of the straight line AB2, and with centre C and distance CD let the circle EFG be described; I.post.3

  • let the straight line EG be bisected at H, I.10 and let the straight lines CG, CH, CE be joined. I.post.1

I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.

For, since GH is equal to HE, and HC is common,

  • the two sides GH, HC are equal to the two sides EH, HC respectively;

and the base CG is equal to the base CE;

  • therefore the angle CHG is equal to the angle EHC. I.8 And they are adjacent angles.

But, 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. I.def.10

Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.

  • Q. E. F.

References

Footnotes

  1. a perpendicular straight line
    , κάθετον εὐθεῖαν γραμμἡν. This is the full expression for a perpendicular, κάθετος meaning let fall or let down, so that the expression corresponds to our plumb-line. ἡ κάθετος is however constantly used alone for a perpendicular, γραμμἡ being understood. ↩

  2. on the other side of the straight line AB
    , literally towards the other parts of the straight line AB, ἐπὶ τὰ ἕτερα μέρη τῆς AB. Cf. on the same side (ἐπὶ τὰ αὐτὰ μέρη) in Post. 5 and in both directions (ἐφ̓ ἑκάτερα τὰ μἑρη) in Def. 23. ↩