Mohr–Mascheroni theorem

The result was originally published by Georg Mohr in 1672,[2] but his proof languished in obscurity until 1928.

Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool.

[7] An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space

[8] It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).

In keeping with the intent of the theorem which we aim to prove, the actual line need not be drawn but for aesthetic reasons.

However, when a construction is being used to prove that something can be done, it is not necessary to describe all these various choices and, for the sake of clarity of exposition, only one variant will be given below.

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.

This equivalence can also be established with (collapsing) compass alone, a proof of which can be found in the main article.

This construction can be repeated as often as necessary to find a point Q so that the length of line segment AQ = n⋅ length of line segment AB for any positive integer n. Point I is such that the radius r of B(r) is to IB as DB is to the radius; or IB / r = r / DB.

In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points),[10] find a point Q on the line BD so that the length of line segment BQ is a positive integral multiple, say n, of the length of BD and is greater than r / 2 (this is possible by Archimede's axiom).

A variation on the compass, a neusis tool which does not actually exist but as an abstraction, has also been studied.

Motivated by Mascheroni's result, in 1822 Jean Victor Poncelet conjectured a variation on the same theme.

Alternatively, a second circle which is neither intersecting nor concentric is sufficient, provided that a point on either the centerline through them or the radical axis between them is given, or two parallel lines exist in the plane.