used and referenced frequently. These are also compass-only constructions. All constructions below rely on #1,#2,#5, and any other construction that is listed prior to it. Compass equivalence theorem (circle translation) The ability to translate, or copy, a circle to a new center is vital in these proofs and fundamental to establishing the veracity of the theorem. The creation of a new circle with the same radius as the first, but centered at a different point, is the key feature distinguishing the collapsing compass from the modern, rigid compass. With the rigid compass this is a triviality, but with the collapsing compass it is a question of construction possibility. The equivalence of a collapsing compass and a rigid compass was proved by Euclid (Book I Proposition 2 of The Elements) using straightedge and collapsing compass when he, essentially, constructs a copy of a circle with a different center. This equivalence can also be established with compass alone, a proof of which can be found in the main article. Reflecting a point across a line Given a line segment and a point not on the line determined by that segment, construct the image of upon reflection across this line. Construct two circles: one centered at and one centered at , both passing through . , the other point of intersection of the two circles, is the reflection of across the line . If (that is, there is a unique point of intersection of the two circles), then is its own reflection and lies on the line (contrary to the assumption), and the two circles are internally tangential. Extending the length of a line segment Given a line segment find a point on the line such that is the midpoint of line segment . Construct point as the intersection of circles and . (∆ABD is an equilateral triangle.) Construct point as the intersection of circles and . (∆DBE is an equilateral triangle.) Finally, construct point as the intersection of circles and . (∆EBC is an equilateral triangle, and the three angles at show that are collinear.) This construction can be repeated as often as necessary to find a point so that the length of line segment = ⋅ length of line segment for any positive integer . Inversion in a circle Given a circle , for some radius (in black) and a point construct the point that is the inverse of in the