三角形的九点二次曲线

The Nine Point Conic to a Triangle

在任意三角形内,三边中点,三高的垂足,以及连接顶点与垂心的三线段的中点,都在同一圆上,此圆即为三角形九点圆.三角形的九点圆是欧氏几何中著名的优美定理,被称为欧拉圆和费尔巴哈圆.本文试图把垂心改换为平面内的任意点,相应地把三条高线改换为过每个顶点各一条的共点直线组时,则将把三角形的九点圆有趣地推广为三角形的九点二次曲线.并具体讨论在不同的区域内得到的九点二次曲线.

In any triangle, the midpoints of three the sides, perpendicular feet, and the midpoints between the orthoeenter （the point where the three altitudes meet） and each of the three vertices, these nine points all lie on a circle. The circle is called Nine Point Circle, and is also called the Euler＇s Circle or the Feuerbaeh Cricle. The Nine Point Circle to a triangle is a famous beautiful theorem in Euclidean Geometry. In this paper, if the orthoeenter is changed into any point in the plane, and the three altitudes are changed into the con-point lines through the three vertexes of triangle, the Nine Point Circle theorem is generalized to nine-point conic theorem.