Geometrical Constructions and Solutions in Mathematics

Explore a variety of geometrical constructions and solutions in mathematics, including elementary algebraic sentences, harmonic division of a segment, the Apollonian problem, and Lemoines Construction. These visualizations illustrate complex and exact methods used in geometric problem-solving, such as constructing parallel lines, dividing segments harmonically, and addressing the Apollonian problem with different solutions.

• Geometrical Constructions
• Mathematics Solutions
• Algebraic Sentences
• Harmonic Division
• Geometric Problems

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1. Geometrical Complexity Lygatsika Ioanna Maria Varvakeios Standard Experimental High School of Athens

2. Elementary algebraic sentences x = a or x > a or x < a

3. Elementary geometrical constructions Place a straightedge s graph edge through a given point R1 R2 Draw a straight line Place a point of a compass on given point Place a point of a compass on an indeterminate point on a line C1 C2 n1R1+n2R2+n 1R 1+m1C1+m2C2+m3C3+ E C3 Draw a circle To place one edge of the square on a given point Displacement of the square from point A along the ruler until you pass from point B R 1 Complexity: n1+n2+n 1+m1+m2+m3+ E Exactitude: n1+ n 1+ m1+ m2

4. Construction of a parallel line Geometrography Construction Construction using square C1+ C3 E 2R1+ R2 R 2 2R 1 C1+ C3 C1+ C3 (2R1+ R2+ 3C1+3C3), 1 line, 3 circles Complexity: 9 (R2 + 2R 1+ E), 1 line Complexity: 4 Exactitude: 3 Exactitude: 5

5. Harmonic division of a segment Geometrography construction Classical construction q p D1 (7R1+ 4R2+ 9C1+ 5C3), 4 lines, 2 circles Complexity: 25 (4R1+ 2R2+ 9C1+ 5C3), 2 lines, 5 circles Complexity: 20 Exactitude: 16 Exactitude: 13

6. The Apollonian Problem Which one of the solutions is the simplest?

7. Lemoines Construction http://www.geogebratube.org/material/show/id/33100 Operation Note R1+R2 1 Line Ax 2 Circles (E, b), (F, b), (E, c), (F, c) 4 *(2C1+C3) 3 *(C1+C3) 2C1+C3 3 Circles (B, ), (B0, ), (B1, ) 4 Circles (A, AL) Circles (B, LB ), (B, LB ), (C, LC ), (C, LC ), (C, LC ) 5 *(3C1+C3) 5 6 Lines OP, O P , O P , IJ, I J , I J 6 *(2R1+R2) 4 *(2R1+R2) 5C1+4C3 16 *(2R1+R2) 7 4 lines AHi 8 4 circles (A, AHi) 9 4 lines H Hi, 4 H Ni, 4 Axi, 4 AX i 104 circles (Di, DiXi), 4 circles (D i, D iX i) 8 *(2C1+C3)

8. Pamfilos Construction http://www.geogebratube.org/material/show/id/33098 Operation Note 3C1 + C3 3C1 + C3 Circle k (A, a+c) Circle l (B, b-c) Points Z and E Line EZ, EC Circle (E, s) Perpendicular Bisector f Circle (L, LC) Line MN 1 2 ( , c) 2R1+R2+6C1+4C3 2 *(2R1+R2) 3C1 + C3 + 6C1 2R1+R2+2C1+2C3 2C1 + C3 2R1+R2 2 *(8R1+4R2+4C1+3C3) 4R1+2R2+5C1+4C3 3C1 + C3 3 4 ( ,r - c) 6 ( , a) 7 8 ( , b) 9 10 Tangents PO and P1O 11 Circle n and centre D b = b - c a = a + c 12 Circle n(D, r-c)

9. Eppsteins Construction http://www.geogebratube.org/material/show/id/33096 Operation through Note Line given points Perpendicular through a given point not on the line through given points two 3 *(2R1+ R2) 1 line 3 *(2R1+R2+3C1+3C3) 2 3Line two 6 *(2R1+ R2) 4Circle through three given points 2 *(4R1+2R2+3C2+3C3)

10. Construction Lemoine Pamfilos Eppstein It is based directly (without a geometrical refinement) on the construction of radical axes (Vieta construction). It is based on reversion, that reduces the construction to an easier method (Soddy circles). It is based on a convenient system of construction of radical axes. Method of construction 27 lines, 25 circles 48 lines, 48 circles 12 lines, 2 circles In total Complexity: 154 Complexity: 808 Complexity: 78 Exactitude: 102 Exactitude: 520 Exactitude: 47

11. More Ideas 1. Study of the statistical model proposed by Mustonen 2. Library of geometrical problems with optimized solutions 3. The role of the Geometric Straight Line Program in the dynamic geometry system 4. Geometrical complexity theory

12. References 1) Pamfilos Paris, ( Elasson Geometrikon ), University of Crete Publications, 2012 2) Mustonen Seppo, Statistical Accurancy http://www.survo.fi/papers/GeomAccuracy.pdf 3) Rudiger Thiele, Hilbert s Twenty-Fourth Problem , THE MATHEMATICAL ASSOCIATION OF AMERICA Monthly 110, January 2003. 4) Lemoine, E., Applications de la geometrographie a l examen de diverses solutions d un meme problem , Bulletin de la S.M.F. tome 20 (1892) p. 132-150. 5) Lemoine, E., Geometrographie , C. Naud, Paris, 1902. 6) Arnaudies J.M. et Delezoide P., Constructions geometriques , Bulletin 446, APMEP. 7) De Temple D. W., Carlyle Circles and the Lemoine Simplicity of Polygon Construtions , The American Mathematical Monthly, vol. 98, issue 2, Fevr. 1991, pp. 97-108 8) Eppstein, David, Tangent Spheres and Triangle Centers , American Mathematical Monthly 108, 63 66, 2001. 9) Gisch, David and J. M. Ribando, Apollonius Problem: A Study of Solutions and Their Connections . American Journal of Undergraduate Research 3(1), 15 25, 2004. 10) Denis Roegel, Kissing circles: A French romance in METAPOST , TUGboat, Volume 26 (2005), No. 1 11) Lemoine s method: http://geometrographie.org/ 12) Epstein s method: http://www.ics.uci.edu/~eppstein/junkyard/tangencies/apollonian.html of Geometric Constructions ,