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.
Read more about this topic: Compass Equivalence Theorem
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