---
book: 1
number: 11
id: "I.11"
kind: "construction"
uses: ["[[book-1/proposition-3]]", "[[book-1/proposition-1]]", "[[book-1/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.11"
license: "CC-BY-SA-4.0"
---

# I.11

To draw a straight line at right angles to a given straight line from a given point on it.

![I.11](figures/I-11.svg)

## Proof

Let *AB* be the given straight line, and *C* the given point on it.

Thus it is required to draw from the point *C* a straight line at right angles to the straight line *AB*.

Let a point *D* be taken at random on *AC*; let *CE* be made equal to *CD*; [[book-1/proposition-3|I. 3]] on *DE* let the equilateral triangle *FDE* be constructed, [[book-1/proposition-1|I. 1]] and let *FC* be joined;

I say that the straight line *FC* has been drawn at right angles to the given straight line *AB* from *C* the given point on it.

For, since *DC* is equal to *CE*, and *CF* is common, the two sides *DC*, *CF* are equal to the two sides *EC*, *CF* respectively; and the base *DF* is equal to the base *FE*; therefore the angle *DCF* is equal to the angle *ECF*; [[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; [[book-1/definitions#Definition 10|Def. 10]] therefore each of the angles *DCF*, *FCE* is right.

Therefore the straight line *CF* has been drawn at right angles to the given straight line *AB* from the given point *C* on it.

Q. E. F.