---
book: 3
number: 34
id: "III.34"
kind: "construction"
uses: ["[[book-1/proposition-23]]", "[[book-3/proposition-32]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.34"
license: "CC-BY-SA-4.0"
---

# III.34

*From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle*.

## Proof

Let *ABC* be the given circle, and the angle at *D* the given rectilineal angle; thus it is required to cut off from the circle *ABC* a segment admitting an angle equal to the given rectilineal angle, the angle at *D*.

Let *EF* be drawn touching *ABC* at the point *B*, and on the straight line *FB*, and at the point *B* on it, let the angle *FBC* be constructed equal to the angle at *D*. [[book-1/proposition-23|I. 23]]

Then, since a straight line *EF* touches the circle *ABC*, and *BC* has been drawn across from the point of contact at *B*, the angle *FBC* is equal to the angle constructed in the alternate segment *BAC*. [[book-3/proposition-32|III. 32]]

But the angle *FBC* is equal to the angle at *D*; therefore the angle in the segment *BAC* is equal to the angle at *D*.

Therefore from the given circle *ABC* the segment *BAC*. has been cut off admitting an angle equal to the given rectilineal angle, the angle at *D*. Q. E. F.