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Mascheroni Construction


A geometric construction done with a movable compass alone. All constructions possible with a compass and straightedge are possible with a movable compass alone, as was proved by Mascheroni (1797). Mascheroni's results are now known to have been anticipated largely by Mohr (1672).

MascheroniMidpoint

An example of a Mascheroni construction of the midpoint M of a line segment specified by two points A and B illustrated above (Steinhaus 1999, Wells 1991). Without loss of generality, take AB=1.

1. Construct circles centered at A and B passing through B and A. These are unit circles centered at (0, 0) and (1, 0).

2. Locate C, the indicated intersection of circles A and B, and draw a circle centered on C passing through points A and B. This circle has center (1/2, sqrt(3)/2) and radius 1.

3. Locate D, the indicated intersection of circles B and C, and draw a circle centered on C passing through points B and C. This circle has center (3/2, sqrt(3)/2) and radius 1.

4. Locate E, the indicated intersection of circles B and D, and draw a circle centered on E passing through point C. This circle has center (2, 0) and radius sqrt(3).

5. Locate F and G, the intersections of circles AE and EC. These points are located at positions (5/4, +/-sqrt(39)/4).

6. Locate M, the intersection of circles F and G. This point has position (1/2, 0), and is therefore the desired midpoint of AB^_.

Pedoe (1995, pp. xviii-xix) also gives a Mascheroni solution.


See also

Compass, Geometric Construction, Neusis Construction, Steiner Construction, Straightedge

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 96-97, 1987.Bogomolny, A. "Geometric Constructions with the Compass Alone." http://www.cut-the-knot.org/do_you_know/compass.shtml.Courant, R. and Robbins, H. "Constructions with Other Tools. Mascheroni Constructions with Compass Alone." §3.5 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 146-158, 1996.Dörrie, H. "Mascheroni's Compass Problem." §33 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 160-164, 1965.Gardner, M. "Mascheroni Constructions." Ch. 17 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 216-231, 1979.Hutt, E. Die Mascheroni'schen Konstruktionen für die zwecke höherer Lehrenstalten und zum Selbstuterrichte. Halle, Germany: H. W. Schmidt, 1880.Mascheroni, L. Geometria del compasso. Pavia, Italy, 1797.Mohr, G. Euclides Danicus. Amsterdam, Netherlands, 1672.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.Quemper de Lanascol, A. Géométrie du compas. Albert Blanchard, pp. 74-77, 1925.Schwerin. Mascheronische Konstruktionen. 1898.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 141-142, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 148-149, 1991.

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Mascheroni Construction

Cite this as:

Weisstein, Eric W. "Mascheroni Construction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MascheroniConstruction.html

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