Mohr–Mascheroni theorem


In mathematics, 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.
It must be understood that by "any geometric construction", we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as:
Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary.

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.
Motivated by Mascheroni's result, in 1822 Jean Victor Poncelet conjectured a variation on the same theme. He proposed that any construction possible by straightedge and compass could be done with straightedge alone. The one stipulation though is that a single circle with its center identified must be provided. The Poncelet-Steiner theorem was proved by Jakob Steiner eleven years later. This was a generalization of the proofs given by Ferrari and Cardano and several others in the 16th century where they demonstrated that all the constructions appearing in Euclid's Elements were possible with a straightedge and a "rusty" compass.

Constructive proof approach

To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a compass alone, as these are the foundations of, or elementary steps for, all other constructions. These are:
  1. Creating the line through two existing points
  2. Creating the circle through one point with centre another point
  3. Creating the point which is the intersection of two existing, non-parallel lines
  4. Creating the one or two points in the intersection of a line and a circle
  5. Creating the one or two points in the intersection of two circles.
#1 - A line through two points
It is understood that a straight line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it. In keeping with the intent of the theorem which we aim to prove, the actual line need not be drawn but for aesthetic reasons. This fact will be shown when all other constructions involving the line are proven.
#2 - A circle through one point with defined center
This can be done with compass alone quite naturally; it is the very purpose for which compasses are meant. There is nothing to prove. Any doubts about this construction would equally apply to traditional constructions that do involve a straightedge.
#5 - Intersection of two circles
This construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle, it can always be assumed that any circle is described by its center and radius. Indeed, some authors include this in their descriptions of the basic constructions.
#3, #4 - The other constructions
Thus, to prove the theorem, there only compass-only constructions for #3 and #4 need to be given.

Alternative proofs

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. This approach can be used to provide a stronger version of the theorem. It also shows the dependence of the theorem on Archimedes' axiom.

Constructive proof

The following notation will be used throughout this article. A circle whose center is located at point and that passes through point will be denoted by. A circle with center and radius specified by a number,, or a line segment will be denoted by or, respectively.
In general constructions there are often several variations that will produce the same result. The choices made in such a variant can be made without loss of generality. 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. However, many constructions come in different forms depending on whether or not they use circle inversion and these alternatives will be given if possible.

Some preliminary constructions

To prove the above constructions #3 and #4, which are included below, a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also compass-only constructions. All constructions below rely on #1,#2,#5, and any other construction that is listed prior to it.

Compass equivalence theorem (circle translation)

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. The creation of a new circle with the same radius as the first, but centered at a different point, is the key feature distinguishing the collapsing compass from the modern, rigid compass. The equivalence of a collapsing compass and a rigid compass was proved by Euclid using straightedge and collapsing compass when he, essentially, constructs a copy of a circle with a different center. This equivalence can also be established with compass alone, a proof of which can be found in the main article.

Reflecting a point across a line

  1. Construct two circles: one centered at and one centered at, both passing through.
  2. , the other point of intersection of the two circles, is the reflection of across the line. If , then lies on the line and equals its own reflection.

    Extending the length of a line segment

  1. Construct point as the intersection of circles and.
  2. Construct point as the intersection of circles and.
  3. Finally, construct point as the intersection of circles and.
This construction can be repeated as often as necessary to find a point so that the length of line segment = ⋅ length of line segment for any positive integer.

Inversion in a circle

  1. Draw a circle .
  2. Assume that the red circle intersects the black circle at and
  3. *if the circles do not intersect in two points see below for an alternative construction.
  4. *if the circles intersect in only one point,, it is possible to invert simply by doubling the length of .
  5. Reflect the circle center across the line :
  6. # Construct two new circles and .
  7. # The light blue circles intersect at and at another point.
  8. Point is the desired inverse of in the black circle.
Point is such that the radius of is to as is to the radius; or.
In the event that the above construction fails, find a point on the line so that the length of line segment is a positive integral multiple, say, of the length of and is greater than . Find the inverse of in circle as above. The point is now obtained by extending so that =.

Determining the center of a circle through three points

  1. Construct point, the inverse of in the circle.
  2. Reflect in the line to the point.
  3. is the inverse of in the circle.

    Intersection of two non-parallel lines (construction #3)

  1. Select circle of arbitrary radius whose center does not lie on either line.
  2. Invert points and in circle to points and respectively.
  3. The line is inverted to the circle passing through, and. Find the center of this circle.
  4. Invert points and in circle to points and respectively.
  5. The line is inverted to the circle passing through, and. Find the center of this circle.
  6. Let be the intersection of circles and.
  7. is the inverse of in the circle.

    Intersection of a line and a circle (construction #4)

The compass-only construction of the intersection points of a line and a circle breaks into two cases depending upon whether the center of the circle is or is not collinear with the line.

Circle center is not collinear with the line

Assume that center of the circle does not lie on the line.
  1. Construct the point, which is the reflection of point across line.
  2. * Under the assumption of this case,.
  3. Construct a circle .
  4. The intersections of circle and the new red circle are points and.
  5. * If the two circles are tangential then.
  6. Points and are the intersection points of circle and the line.
  7. * If then the line is tangential to the circle.
An alternate construction, using circle inversion can also be given.
  1. Invert points and in circle to points and respectively.
  2. The line is inverted to the circle passing through, and. Find the center of this circle.
  3. and are the intersection points of circles and.

    Circle center is collinear with the line

  1. Construct point as the other intersection of circles and.
  2. Construct point as the intersection of circles and.
  3. Construct point as the intersection of circles and.
  4. Construct point as an intersection of circles and.
  5. Points and are the intersections of circles and.
Thus it has been shown that all of the basic construction one can perform with a straightedge and compass can be done with a compass alone, provided that it is understood that a line cannot be literally drawn but merely defined by two points.