无尺作图

Graphing without straightedge

探讨几何作图体系中直尺的作用。结论是：在欧氏几何作图问题（包括纯作图命题与其他命题的作图部分）中，凡是用圆规和直尺能完成的作图问题，都能只用圆规完成（无尺作图）。首先给出无尺作图中直线、线段、射线等的作图约定。其次给出一个无尺作图的基础作图体系，为论证结论提供手段。再次给出直尺在几何作图中的６个基本操作，以及无尺作图时相应的６个替代操作；使用替代操作就能把任何有解的几何作图问题和它的求解，一同“翻译”成相应的无尺作图问题及其求解，从而导出结论；最后指出简化此“翻译”的方法，并以此方法创立一门“无尺几何”的途径。

The functions of straightedge in Euclidean geometry are explored. The conclusion is: In Euclidean geometry, the graphing problems, including pure graphing propositions or graphing parts of other propositions, which can be solved with compasses and straightedge can actually be done by compasses only (i.e., graphing without straightedge). First, some stipulations in the context of graphing without straightedge are presented. These stipulations include that of straight line, line segment, ray, and basic graphs composed by them. Second, the fundamental system of graphing without straightedge is established. This graphing system serves as basic approach to prove the conclusion. Third, six basic operations employing straightedge in geometry graphing and their correspondent substitutes in the graphing without straightedge are exhibited firstly. Then it is proved that all solvable graphing problems of geometry and their solutions can 'be translated' into correspondent problems and solutions in the graphing without straightedge, where our conclusion is reached. Lastly, the method to simplify the 'translation' procedure and the way to initiate a 'geometry without straightedge' with this method are presented.