This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. plangeo1 http://www.openmath.org/cd http://www.win.tue.nl/~amc/oz/om/cds/plangeo1.ocd 2006-06-01 experimental 2004-06-01 0 5 Author: Arjeh Cohen This CD defines symbols for planar Euclidean geometry. point The symbol is used to indicate a point of planar Euclidean geometry by a variable. The point may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments. Given two lines l and m, a point A on l and m is defined by: line The symbol is used to indicate a line of planar Euclidean geometry by a variable. The line may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments. Given points A and B, a line l through A and B is defined by: incident The symbol represents the logical incidence function which is a binary function taking arguments representing geometric objects like points and lines and returning a boolean value. It is true if and only if the first argument is incident to the second. That a point A is incident to a line l is given by: configuration The symbol represents a configuration in Euclidean planar geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations. The configuration of a point A and a line l incident to A is defined by: The prevous configuration of a point A and a line l incident with A can be extended by adding a second point B incident with l: We describe a triangle on the distinct points A, B, C and lines a, b, c: type The symbol represents the type of the basic geometric objects: points, lines, configuration. If A and B are objects of the same type, then they are not incident. assertion The symbol is a constructor with two arguments. Its first argument should be a configuration, its second argument a statement about the configuration, called thesis. When applied to a configuration C and a thesis T, the OpenMath object assertion(C,T) expresses the assertion that T holds in C. The assertion that two distinct lines meet in only one point can be expressed as follows using the assertion symbol. are_on_line The statement that a set of points is collinear. This example states that A, B, C, and D are collinear.