In Euclidean geometry, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. This theorem refers to geometric constructions which only involve points and circles, since it is not possible to draw straight lines without a straightedge. However, a line is considered to be determined if two distinct points on that line are given or constructed, even if the line itself is not drawn. Although the use of a straightedge can make certain constructions significantly easier, the theorem shows that these constructions are possible even without the use of it. This means the only use of a straightedge is for the aesthetics of drawing straight lines, and is functionally unnecessary for the purposes of construction.
History The result was originally published by Georg Mohr in 1672, but his proof languished in obscurity until 1928. The theorem was independently discovered by Lorenzo Mascheroni in 1797 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered. Several proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool. Mohr's solution was different. In 1890, August Adler published a proof using the inversion transformation. An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space R 2 {\displaystyle \mathbb {R} ^{2}} . In this way, a stronger version of the theorem was proven in 1990. It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
Constructive proof
Outline To prove the Mohr–Mascheroni theorem, it suffices to show that each of the basic constructions of compass and straightedge is possible using a compass alone, as these are the foundations of all other constructions. All constructions can be written as a series of steps involving these five basic constructions:
Creating the line through two existing points Creating the circle through one point with centre another point Creating the point which is the intersection of two existing, non-parallel lines Creating the one or two points in the intersection of a line and a circle (if they intersect) Creating the one or two points in the intersection of two circles (if they intersect). Constructions (2) and (5) can be done with a compass alone. For construction (1), a line is considered to be given by any two points. It is understood that the line itself cannot be drawn without a straightedge, so the proof of the theorem lies in showing that constructions (3) and (4) are possible using only a compass. Once this is done, it follows that every compass-straightedge construction can be done under the restrictions of the theorem.
Notation The following notation will be used throughout this article. A circle whose center is located at point U and that passes through point V will be denoted by U(V). A circle with center U and radius specified by a number, r, or a line segment AB will be denoted by U(r) or U(AB), respectively.
Some preliminary constructions To prove the above constructions (3) and (4), a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also compass-only constructions.
Compass equivalence theorem (circle translation)
The modern compass with its fixable (non-collapsing) aperture can be used to transfer distances directly, while a collapsing compass cannot. The compass equivalence theorem states that, while a "modern compass" appears to be a more powerful instrument, it can be simulated with a collapsing compass alone. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) for the proof of this theorem. Given points A, B, and C, construct a circle centered at A with the radius BC, using only a collapsing compass.
Draw a circle centered at A and passing through B and vice versa (the blue circles). They will intersect at points D and D'. Draw circles through C with centers at D and D' (the red circles). Label their other intersection E. Draw a circle (the green circle) with center A passing through E. This is the required circle.
Reflecting a point across a line
Given a line AB determined by two points A and B, and an arbitrary point C, construct the image of C upon reflection across this line:
Construct two circles: one centered at A and one centered at B, both passing through C. The other point of intersection of the two circles, D, is the reflection of C across the line AB. If C = D (that is, there is a unique point of intersection of the two circles), then C is its own reflection and lies on the line AB.
Extending the length of a line segment
Given a line AB determined by two points A and B, construct the point C on the line such that B is the midpoint of line segment AC.
Construct point D as the intersection of circles A(B) and B(A). (∆ABD is an equilateral triangle.) Construct point E ≠ A as the intersection of circles D(B) and B(D). (∆DBE is an equilateral triangle.) Finally, construct point C ≠ D as the intersection of circles B(E) and E(B). (∆EBC is an equilateral triangle, and the three angles at B show that A, B and C are collinear.) This construction can be repeated as often as necessary to find a point Q so that the length of line segment AQ is n times the length of line segment AB for any positive integer n.
Inversion in a circle
Given a circle B(r), for some radius r (in black) and a point D (≠ B), construct the point I that is the inverse of D about the circle. Naturally there is no inversion for a point D = B.
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