Mohr–Mascheroni theorem

The set of Mohr- Mascheroni from the synthetic geometry states that any structure that can be done with ruler and compass, is already using compasses alone possible. It is named after the mathematicians Georg Mohr and Lorenzo Mascheroni that it proved independently.

History

Was first proved the theorem in 1672 by Georg Mohr. His proof came but forgotten, so that the sentence was in 1797 again proved by Lorenzo Mascheroni. Only later, Mohr was evidence rediscovered by Johannes Hjelmslev, and named the set after the two mathematicians. This was followed by a series of much simpler proofs. Most of these proofs are elementary geometric, by Jean -Claude Carrega comes an algebraic proof of the statement.

Statement

The set states the following: If there is - starting from a given set of points - one method of construction, which has a point P constructed by the repeated application of the basic structures 1 to 5, so there is also a method which constructs the point P from the same starting position, but here only the structures 2 and 5 used. The elementary structures are:

Idea of ​​proof

Most of the evidence boil down to specify construction methods, such as the intersection of two lines or the intersections of a line and a circle with a compass alone can be determined, the lines are given by only two points. The following proof idea makes use of the inversion at the circuit advantage and goes back to August eagle.

The inversion in the circle has the property that it maps lines and circles that do not go through the center of inversion to channels. Thus, the intersection point may be attributed to the intersection of two circles, which can be constructed directly.

For the intersection of two lines you have it do the following: Given four points A, B, C and D, looking for the point of intersection S of the line AB with the line CD. First, choose an arbitrary point O, which may be situated on any of the lines, and draw any circle with O as center. Is then constructed, the pixels A ', B ', C 'and D' of the four given point in inversion in this circuit and the circles through the points A ', B' and O, as well as C ', D' and O. The two circles are the images of the lines AB and CD intersect and except in O in a further point S ', the pixel of the desired intersection. The intersection S is obtained from this by re- inversion.

Analogously one proceeds from line and circle in the case of the cut. It is therefore sufficient, dots, straight lines and circular mirror with compass alone at a given circle, to which constructions are known. For the construction of the image of a single point is explained in Circle Reflection # With compass alone.

Related phrases

While the set of Mohr- Mascheroni says that the ruler is superfluous, can not be waived on the circle. However, can be prepared by the set of all Poncelet -Steiner constructions perform alone with ruler, if a circuit including the center is given. Instead of the circle with center and two intersecting circles, or any three circles are sufficient, of which the center does not have to be known.