Mohr–Mascheroni theorem

Template:Short description 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 "any geometric construction" refers 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:<ref>Template:Harvnb</ref>

Any Euclidean construction, insofar as the given and required elements are points (or circles), may be completed with the compass alone if it can be completed with both the compass and the straightedge together.

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,<ref>Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).</ref> but his proof languished in obscurity until 1928.<ref name=Eves199>Template:Harvnb</ref><ref>Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B, pages 1–7.</ref><ref>Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A, pages 34–36.</ref> The theorem was independently discovered by Lorenzo Mascheroni in 1797 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered.<ref>Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.</ref>

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.<ref name="Eves199" /> In 1890, August Adler published a proof using the inversion transformation.<ref>Template:Harvnb</ref>

An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space <math>\mathbb{R}^2</math>. In this way, a stronger version of the theorem was proven in 1990.<ref>Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.</ref> 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 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 (if they intersect)
5. Creating the one or two points in the intersection of two circles (if they intersect).

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

#2 - A circle through one point with defined center

This can be done with a compass alone. A straightedge is not required for this.

#5 - Intersection of two circles

This construction can also be done directly with a compass.

#3, #4 - The other constructions

Thus, to prove the theorem, only compass-only constructions for #3 and #4 need to be given.

Notation and remarks

The following notation will be used throughout this article. A circle whose center is located at point Template:Mvar and that passes through point Template:Mvar will be denoted by Template:Math. A circle with center Template:Mvar and radius specified by a number, Template:Mvar, or a line segment Template:Math will be denoted by Template:Math or Template:Math, respectively.<ref>Template:Harvnb</ref>

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.

It is also important to note that some of the constructions below proving the Mohr–Mascheroni theorem require the arbitrary placement of points in space, such as finding the center of a circle when not already provided (see construction below). In some construction paradigms - such as in the geometric definition of the constructible number - the arbitrary placement of points may be prohibited. In such a paradigm, however, for example, various constructions exist so that arbitrary point placement is unnecessary. It is also worth pointing out that no circle could be constructed without the compass, thus there is no reason in practice for a center point not to exist.

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)

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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. With the rigid compass this is a triviality, but with the collapsing compass it is a question of construction possibility. The equivalence of a collapsing compass and a rigid compass was proved by Euclid (Book I Proposition 2 of The Elements) 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 (collapsing) compass alone, a proof of which can be found in the main article.

Reflecting a point across a line

File:PointReflection.png

Point symmetry

• Given a line segment Template:Math and a point Template:Mvar not on the line determined by that segment, construct the image of Template:Mvar upon reflection across this line.

1. Construct two circles: one centered at Template:Mvar and one centered at Template:Mvar, both passing through Template:Mvar.
2. Template:Mvar, the other point of intersection of the two circles, is the reflection of Template:Mvar across the line Template:Math.
   • If Template:Math (that is, there is a unique point of intersection of the two circles), then Template:Mvar is its own reflection and lies on the line Template:Math (contrary to the assumption), and the two circles are internally tangential.

Extending the length of a line segment

File:Compass only extension of a segment.svg

A compass-only construction of doubling the length of segment AB

• Given a line segment Template:Math find a point Template:Mvar on the line Template:Math such that Template:Mvar is the midpoint of line segment Template:Math.<ref name="Pedoe 1988 loc=p. 78">Template:Harvnb</ref>

1. Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (∆ABD is an equilateral triangle.)
2. Construct point Template:Math as the intersection of circles Template:Math and Template:Math. (∆DBE is an equilateral triangle.)
3. Finally, construct point Template:Math as the intersection of circles Template:Math and Template:Math. (∆EBC is an equilateral triangle, and the three angles at Template:Mvar show that Template:Math are collinear.)

This construction can be repeated as often as necessary to find a point Template:Mvar so that the length of line segment Template:Math = Template:Math⋅ length of line segment Template:Math for any positive integer Template:Math.

Inversion in a circle

File:InversionPointCircle.png

Point inversion in a circle

• Given a circle Template:Math, for some radius Template:Mvar (in black) and a point Template:Math construct the point Template:Mvar that is the inverse of Template:Mvar in the circle.<ref>Template:Harvnb</ref> Naturally there is no inversion for a point <math display="inline">D=B</math>.

1. Draw a circle Template:Math (in red).
2. Assume that the red circle intersects the black circle at Template:Mvar and Template:Mvar
   • if the circles do not intersect in two points see below for an alternative construction.
   • if the circles intersect in only one point, <math>E=E'</math>, it is possible to invert <math>D</math> simply by doubling the length of <math>EB</math> (quadrupling the length of <math>DB</math>).
3. Reflect the circle center <math>B</math> across the line <math>EE'</math>:
   1. Construct two new circles Template:Math and Template:Math (in light blue).
   2. The light blue circles intersect at Template:Mvar and at another point Template:Math.
4. Point Template:Mvar is the desired inverse of Template:Mvar in the black circle.

Point Template:Mvar is such that the radius Template:Mvar of Template:Math is to Template:Mvar as Template:Mvar is to the radius; or Template:Math.

In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points),<ref name="Pedoe 1988 loc=p. 78"/> find a point Template:Mvar on the line Template:Math so that the length of line segment Template:Math is a positive integral multiple, say Template:Mvar, of the length of Template:Math and is greater than Template:Math (this is possible by Archimede's axiom). Find Template:Mvar the inverse of Template:Mvar in circle Template:Math as above (the red and black circles must now intersect in two points). The point Template:Mvar is now obtained by extending Template:Math so that Template:Math = Template:Math.

Determining the center of a circle through three points

File:Circle center construction.svg

Compass-only construction of the center of a circle through three points (A, B, C)

• Given three non-collinear points Template:Mvar, Template:Mvar and Template:Mvar, find the center Template:Mvar of the circle they determine.<ref name=Pedoe123>Template:Harvnb</ref>

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

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

File:Line intersection by compass.svg

Compass-only construction of the intersection of two lines (not all construction steps shown)

• Given non-parallel lines Template:Math and Template:Math, find their point of intersection, Template:Mvar.<ref name=Pedoe123 />

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

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.

File:LineCircleIntersection.png

Line-circle intersection (non-collinear case)

• Given a circle Template:Math (in black) and a line Template:Math. We wish to construct the points of intersection, Template:Mvar and Template:Mvar, between them (if they exist).<ref name=Hunger784>Template:Harvnb</ref><ref name="Eves199"/>

1. Construct the point Template:Mvar, which is the reflection of point Template:Mvar across line Template:Math. (See above.)
   • Under the assumption of this case, Template:Math.
2. Construct a circle Template:Math (in red). (See above, compass equivalence.)
3. The intersections of circle Template:Math and the new red circle Template:Math are points Template:Mvar and Template:Mvar.
   • If the two circles are (externally) tangential then <math>P=Q</math>.
     • Internal tangency is not possible.
   • If the two circles do not intersect then neither does the circle with the line.
4. Points Template:Mvar and Template:Mvar are the intersection points of circle Template:Math and the line Template:Math.
   • If <math>P=Q</math> then the line is tangential to the circle <math>C(r)</math>.

An alternate construction, using circle inversion can also be given.<ref name=Pedoe123 />

• Given a circle Template:Math and a line Template:Math. We wish to construct the points of intersection, Template:Mvar and Template:Mvar, between them (if they exist).

1. Invert points Template:Mvar and Template:Mvar in circle Template:Math to points Template:Mvar and Template:Mvar respectively.
   • Under the assumption of this case, points Template:Mvar, Template:Mvar, and Template:Mvar are not collinear.
2. Find the center Template:Mvar of the circle passing through points Template:Mvar, Template:Mvar, and Template:Mvar.
3. Construct circle Template:Math, which represents the inversion of the line Template:Math into circle Template:Math.
4. Template:Mvar and Template:Mvar are the intersection points of circles Template:Math and Template:Math.<ref>Pedoe carries out one more inversion at this point, but the points Template:Mvar and Template:Mvar are on the circle of inversion and so are invariant under this last unneeded inversion.</ref>
   • If the two circles are (internally) tangential then <math>P=Q</math>, and the line is also tangential.

Circle center is collinear with the line

File:Circle line intersection.svg

Compass only construction of intersection of a circle and a line (circle center on line)

• Given the circle Template:Math whose center Template:Mvar lies on the line Template:Math, find the points Template:Mvar and Template:Mvar, the intersection points of the circle and the line.<ref>Template:Harvnb</ref>

1. Construct point Template:Math as the other intersection of circles Template:Math and Template:Math.
2. Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (Template:Mvar is the fourth vertex of parallelogram Template:Mvar.)
3. Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (Template:Mvar is the fourth vertex of parallelogram Template:Mvar.)
4. Construct point Template:Mvar as an intersection of circles Template:Math and Template:Math. (Template:Mvar lies on Template:Math.)
5. Points Template:Mvar and Template:Mvar are the intersections of circles Template:Math and Template:Math.

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.

Other types of restricted construction

Restrictions involving the compass

Renaissance mathematicians Lodovico Ferrari, Gerolamo Cardano and Niccolò Fontana Tartaglia and others were able to show in the 16th century that any ruler-and-compass construction could be accomplished with a straightedge and a fixed-width compass (i.e. a rusty compass).<ref>Template:Citation</ref>

The compass equivalency theorem shows that in all the constructions mentioned above, the familiar modern compass with its fixable aperture, which can be used to transfer distances, may be replaced with a "collapsible compass", a compass that collapses whenever it is lifted from a page, so that it may not be directly used to transfer distances. Indeed, Euclid's original constructions use a collapsible compass. It is possible to translate any circle in the plane with a collapsing compass using no more than three additional applications of the compass over that of a rigid compass.

A variation on the compass, a neusis tool which does not actually exist but as an abstraction, has also been studied. Known as the cyclos, the device draws circles similarly to the compass, but does so not by defining a radius or providing a center, but by two points defining a diameter, or by three non-collinear points defining the arc. In either case, a single application of the tool is used, by definition, to draw a complete circle. The cyclos tool has been shown to be equivalent to a compass.

Restrictions excluding the compass

Motivated by Mascheroni's result, in 1822 Jean Victor Poncelet conjectured a variation on the same theme. His work paved the way for the field of projective geometry, wherein he proposed that any construction possible by straightedge and compass could be done with straightedge alone. However, the one stipulation is that no less than a single circle with its center identified must be provided. This statement, now known as the Poncelet–Steiner theorem, was proved by Jakob Steiner eleven years later.

A proof later provided in 1904 by Francesco Severi relaxes the requirement that one full circle be provided, and shows that any small arc of the circle, so long as the center is still provided, is still sufficient.<ref>Template:Harvnb</ref>

Additionally, the center itself may be omitted instead of portions of the arc, if it is substituted for something else sufficient, such as a second concentric circle, a second intersecting circle, or a third circle in the plane. 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. A single circle without its center can also be sufficient under the right circumstances. Other unique conditions may exist.

See also

• Napoleon's problem
• Geometrography
• Inversive geometry
• Projective geometry

References

• Template:Citation
• Template:Citation
• Template:Citation

Further reading

• Template:Citation
• Template:Citation

External links

• Construction with the Compass Only