Proposition I-3

Proposition I-3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
Proof.

Since the length of C is the same as the length of the segment AD, and since that length is the same as AE, it follows that the orginal segment AB has been cut off, with one part equal to the smaller segment.

QED

Given the two unequal segments AB and C, with AB the larger segment and C the smaller segment.

Construct a segment at A which has the length of C.

The segment AD has the length of C.

Now construct a circle with center A and radius the length of AD.

Label this circle F.

Label the point E where the circle F and the segment AB intersect.

QED