---
book: 3
number: 25
id: "III.25"
kind: "construction"
uses: ["[[book-1/proposition-6]]", "[[book-3/proposition-9]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.25"
license: "CC-BY-SA-4.0"
---

# III.25

*Given a segment of a circle*, *to describe the complete circle of which it is a segment*.

## Proof

Let *ABC* be the given segment of a circle; thus it is required to describe the complete circle belonging to the segment *ABC*, that is, of which it is a segment.

For let *AC* be bisected at *D*, let *DB* be drawn from the point *D* at right angles to *AC*, and let *AB*. be joined; the angle *ABD* is then greater than, equal to, or less than the angle *BAD*.

First let it be greater; and on the straight line *BA*, and at the point *A* on it, let the angle *BAE* be constructed equal to the angle *ABD*; let *DB* be drawn through to *E*, and let *EC* be joined.

Then, since the angle *ABE* is equal to the angle *BAE*, the straight line *EB* is also equal to *EA*. [[book-1/proposition-6|I. 6]]

And, since *AD* is equal to *DC*, and *DE* is common, the two sides *AD*, *DE* are equal to the two sides *CD*, *DE* respectively; and the angle *ADE* is equal to the angle *CDE*, for each is right; therefore the base *AE* is equal to the base *CE*.

But *AE* was proved equal to *BE*; therefore *BE* is also equal to *CE*; therefore the three straight lines *AE*, *EB*, *EC* are equal to one another.

Therefore the circle drawn with centre *E* and distance one of the straight lines *AE*, *EB*, *EC* will also pass through the remaining points and will have been completed. [[book-3/proposition-9|III. 9]]

Therefore, given a segment of a circle, the complete circle has been described.

And it is manifest that the segment *ABC* is less than a semicircle, because the centre *E* happens to be outside it.

Similarly, even if the angle *ABD* be equal to the angle *BAD*, *AD* being equal to each of the two *BD*, *DC*, the three straight lines *DA*, *DB*, *DC* will be equal to one another, *D* will be the centre of the completed circle, and *ABC* will clearly be a semicircle.

But, if the angle *ABD* be less than the angle *BAD*, and if we construct, on the straight line *BA* and at the point *A* on it, an angle equal to the angle *ABD*, the centre will fall on *DB* within the segment *ABC*, and the segment *ABC* will clearly be greater than a semicircle.

Therefore, given a segment of a circle, the complete circle has been described. Q. E. F.