|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.||I.3|
|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.||IV.Def.7|
This proposition is used in the proofs of IV.10, IV.16, and occasionally in Books X, XI, and XII.
Next proposition: IV.2