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

Let *ABC* be the given segment of a circle.

It is required to describe the complete circle belonging to the segment *ABC,* that is, of which it is a segment.

Bisect *AC* at *D,* draw *DB* from the point *D* at right angles to *AC,* and join *AB.*

The angle *ABD* is then greater than, equal to, or less than the angle *BAD.*

First let it be greater. Construct the angle *BAE* on the straight line *BA,* and at the point *A* on it, equal to the angle *ABD.* Draw *DB* through to *E,* and join *EC.*

Then, since the angle *ABE* equals the angle *BAE,* the straight line *EB* also equals *EA.*

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

But *AE* was proved equal to *BE,* therefore be also equals *CE.* Therefore the three straight lines *AE, EB,* and *EC* equal one another.

Therefore the circle drawn with center *E* and radius one of the straight lines *AE, EB,* or *EC* also passes through the remaining points and has been completed.

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 center *E* happens to be outside it.

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

But, if the angle *ABD* is 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 center 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.