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

Let *ABC* be the given circle, and the angle at *D* the given rectilinear angle.

It is required to cut off from the circle *ABC* a segment admitting an angle equal to the given rectilinear angle, the angle at *D.*

Draw *EF* touching *ABC* at the point *B.* Construct the angle *FBC* equal to the angle at *D* on the straight line *FB* and at the point *B* on it.

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* equals the angle constructed in the alternate segment *BAC.*

But the angle *FBC* equals the angle at *D,* therefore the angle in the segment *BAC* equals the angle at *D.*

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

Q.E.F.

This proposition is not used in the rest of the *Elements.*