Compass Equivalence Theorem - Alternative Construction Without Straightedge

Alternative Construction Without Straightedge

It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves in proofs of the Mohr-Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.

We are given points A, B, and C, and wish to construct a circle centered at A with the same radius as BC, using only a collapsing compass and no straightedge.

• Draw a circle centered at A and passing through B and vice versa (the blue circles). They will intersect at points D and D'.
• Now draw circles through C with centers at D and D' (the red circles). Find their other intersection and label it E.
• Draw a circle (the green circle) with center A passing through E.

• The line DD' is the perpendicular bisector of AB. Thus A is the reflection of B through line DD'.
• By construction, E is the reflection of C through line DD'.
• Since reflection is an isometry, it follows that AE=BC as desired.