An Alternative to Euclid's Collapsible Compass Construction
An Alternative to Euclid's Collapsing Compass Construction:
Given : Segment AB and Point C
Construct : A segment onto Point C congruent to Segment AB 

1. Construct a Ray starting at Point B and that goes through the random Point C. 

2. Construct at circle with center at Point B and circumference through Point A.
This will create a new point D. 

3. Construct a circle with center at the new point D and circumference through Point B.
This will create a new point E. 

4. Construct a circle with center at the original random point C and the circumference through Point B.
This will create a new point F. 

5. Construct the midpoint of Segment EF. (The construction for the midpoint has not been shown here to avoid clutter.)
Result : Segment GC will be congruent the original given Segment AB. 