Book 1 - Proposition 3
To cut off from the greater of two given unequal straight lines a straight line equal to the less.


Let line AB and line CZ be the two given unequal straight lines, and let line AB be the greater of them.
It is required to cut off from line AB the greater a straight line equal to line CZ the less.

Place line AD at the point A equal to the straight line CZ,

and describe the circle DEF with center A and radius AD.

Now, since the point A is the center of the circle DEF, therefore AE equals AD.

But CZ also equals AD, therefore each of the straight lines AE and CZ equals AD, so that AE also equals CZ.

Therefore, given the two straight lines AB and CZ, AE has been cut off from AB the greater equal to CZ the less.