Selected Topics in Geometry with Classical Vs. Computer ProvingThis textbook presents various automatic techniques based on Gr bner bases elimination to prove well-known geometrical theorems and formulas. Besides proving theorems, these methods are used to discover new formulas, solve geometric inequalities, and construct objects ? which cannot be easily done with a ruler and compass.Each problem is firstly solved by an automatic theorem proving method. Secondly, problems are solved classically ? without using computer where possible ? so that readers can compare the strengths and weaknesses of both approaches. |
Contents
1 Introduction | 1 |
2 Automatic theorem proving | 5 |
3 Generalization of the formula of Heron | 33 |
4 SimsonWallace theorem | 65 |
5 Transversals in a polygon | 91 |
6 PetrDouglasNeumann theorem | 127 |
7 Geometric inequalities | 163 |
8 Regular polygons | 187 |
9 Miscellaneous | 203 |
Conclusion | 225 |
227 | |
235 | |
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Common terms and phrases
affine affine-regular algebraic equations angle arbitrary point area principle areas of triangles automatic theorem proving Brahmagupta Cartesian coordinate system Cartesian system centroid Ceva’s theorem Choose a Cartesian circumcenter circumcircle circumcircle of ABC circumradius classical proof collinear complanar conclusion polynomial convex cubic surface cyclic pentagon cyclic quadrilateral Denote diagonals elimination ideal Elimination of variables equality Euler’s formula of Heron Geometry with Classical given Gröbner basis heptagon holds intersect isoperimetric inequality isosceles triangles KLMN Menelaus method n-gon Napoleon’s theorem non-convex non-degeneracy conditions normal form parallelogram PDN theorem planar plane problem quadrilateral ABCD radical ratios of points regular pentagon Regular polygons relation respectively Selected Topics side lengths sides BC signed area Similarly Simson–Wallace theorem skew quadrilateral slack variable solve statement sum of squares system of coordinates tetrahedron Thébault's theorem Topics in Geometry triangle ABC vector vertex vertices