射影几何中抛物线的若干问题

Some Parabola-related Problems in Projective Geometry

在射影几何中,作为二级曲线的抛物线可以看作由定点与定直线上的动点连线的中垂线构成.二阶曲线若存在一个外切三角形,其外接圆过二阶曲线的焦点,则该二阶曲线为抛物线.过定点的二阶曲线的三条弦,若每条弦的两端点处切线正交,则二阶曲线为抛物线.给定三角形外接圆上任一点(不是顶点),存在唯一抛物线以给定点为焦点,与已知三角形三边相切.

In the projective geometry, as a curve of order 2, the parabolas may be regarded as consisting of a straight line, which itself is a perpendicular bisector of lines formed by joining fixed-points and the moving points on a fixed straight line, if a curve of order 2 has a circumscribed triangle, and its circumcircle is right on its focal point, then this curve of order 2 is a parabola; of the three subtenses of a curve of order 2 which passes a fixed point, if the two tangent lines on the two end points of each of these three subtenses are orthogonal, then such a curve of order 2 must be a parabola. On any given point （except the apices） of the circumcircle of a triangle, there exists an only parabola whose focal point lies in a given point and at the same time it is tangental with the three sides of the given triangle.