Compass equivalence theorem

From Wikipedia, the free encyclopedia

The compass equivalence theorem is a statement of compass and straightedge constructions that whenever the compass is lifted from the page, it is assumed to collapse, so may not be directly used to transfer distances.

This might seem a difficult obstacle to surmount, but it is not. Any construction via a "fixed" compass may be attained with a collapsing compass. It is possible to construct a circle of equal radius, centered at any given point on the plane.

[edit] Proof

A proof by construction will follow. Suppose we have a circle o centered at A, and we want to construct a congruent circle centered at a given point B.

• Draw circles d₁ and d₂, centered at A and passing through B and vice versa, respectively.
• Let the intersection of d₁ and d₂ be C, and let the intersection of o and d₂ be D.
• Construct the circle f centered at C, passing through D.
• Let F be the intersection of f and d₁ which is not on the same side of the bisector of the segment AB as D.
• The circle centered at B, passing through F is symmetrical to o, and thus congruent, which was to be found.

[edit] External links

• Minnesota State University Mathematics Department