To fit a straight line into a given circle equal to a given straight line which is not greater than the diameter of the circle.

Let *ABC* be the given circle, and *D* the given straight line not greater than the diameter of the circle

It is required to fit a straight line into the circle *ABC* equal to the straight line *D.*

Draw a diameter *BC* of the circle *ABC.*

If *BC* equals *D,* then that which was proposed is done, for *BC* has been fitted into the circle *ABC* equal to the straight line *D.*

But, if *BC* is greater than *D,* make *CE* equal to *D,* describe the circle *EAF* with center *C* and radius *CE,* and join *CA.*

Then, since the point *C* is the center of the circle *EAF, CA* equals *CE.*

But *CE* equals *D,* therefore *D* also equals *CA.*

Therefore *CA* has been fitted into the given circle *ABC* equal to the given straight line *D.*

Q.E.F.

This proposition is used in the proofs of IV.10, IV.16, and occasionally in Books X, XI, and XII.