The proofs produced by Euclid are elegant, short (often shorter than the proofs given by geometers) and understandable even to high school students. This method seems to be the first that can produce traditional proofs for hard geometry theorems automatically....
In this section we prove Theorems 4.2 and 4.3 by constructing a sequence of [Math Processing Error]ε-approximate Steiner trees [Math Processing Error]Tk ([Math Processing Error]k∈N), for which it is possible to calculate the ratio between their length and the length of a locally minimum ...
The other two equations, (80) and (81), are proved analogously. 6. Concluding Remarks We have proved three hyperbolic cosine-sine theorems for the triangular formed by arcs of three intersecting semicircles by using only elements of the Euclidean geometry. This geometrical figure is one of the...
Beitr¨age zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002),No. 1, 27-31. Con,guration Spaces of Weighted Graphs in High Dimensional Euclidean Spaces Olivier Mermoud Marcel Steiner D´partement de G´nie M´canique, ICAP-LICP,EPFL CH-1015 Lausanne e-mail...
Yang L,Gao X S,Chou S C,et al.Automated production of readable proofs for theorems in non-Euclidean geometries[A].Automated deduction in geometry.Lecture notes in artificial intelligence 1360[C].Berlin:Springer-Verlag,1997.171—188.Yang L , Gao X S , Chou S C , et al. Automated ...
However, it will be useful in the proofs of the theorems that follow. The process that constructs the metric g2 from g1 can then be repeated to create a third metric g3, and so on. This generates a sequence of metrics (gi ), obtained from a sequence of functions (hi ) (Fig. 3)...
Euclidean geometry is the study of plane and solid figures on the basis of axioms and theorems employed by the ancient Greek mathematician Euclid. The term refers to the plane and solid geometry commonly taught in secondary school. Euclidean geometry is
have found more than 300 distinct proofs of the Pythagorean theorem. Despite its antiquity, it remains one of the most important theorems inmathematics. It enables one to calculate distances or, more important, to define distances in situations far more general than elementary geometry. For example...