三角二次曲线和三次曲线
Triangle conics and cubics
摘要：
本文用射影几何的基本原理讨论了经典几何中的三角形、三次曲线和二次曲线。近年来，N.J. Wildberger使用代数的观点积极地处理这个话题。H. M. Cundy和C.F.帕里最近本文的主要任务是开发一种算法，用于创建通过三角形中心的曲线。在研究过程中，注意到不同三角形中的一些不同三角形中心重合。最简单的例子：一个底三角形的一个圆心是一个外心三角形的一个正交心。这是创建算法的关键。实际上，我们可以将属于一条曲线（基曲线）的点与另一个三角形的其他点匹配。从而得到一个新的有趣的几何对象。在研究过程中，得到了一些新的三角二次曲线和三次曲线，并讨论了它们在欧氏空间中的性质。此外，还讨论了所得到的定理在射影几何中的推论，证明了所发现的结果都可以推广到射影平面上。
英文摘要：
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.
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