---
book: 1
number: 12
id: "I.12"
kind: "construction"
uses: ["[[book-1/proposition-10]]", "[[book-1/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.12"
license: "CC-BY-SA-4.0"
---

# I.12

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

![I.12](figures/I-12.svg)

## Proof

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 *AB*, and with centre *C* and distance *CD* let the circle *EFG* be described; [[book-1/postulates#Postulate 3|Post. 3]] let the straight line *EG* be bisected at *H*, [[book-1/proposition-10|I. 10]] and let the straight lines *CG*, *CH*, *CE* be joined. [[book-1/postulates#Postulate 1|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*. [[book-1/proposition-8|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. [[book-1/definitions#Definition 10|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.