Research On Special Centers Of Triangle

Triangle is the most basic plane figure,and the theory of plane geometry about triangle is the most mature?The special point of triangle has many wonderful properties.Like human eyes,it is the window of triangle "mind"?The most well-known special points are the Five centers(barycenter,excenter,incenter,orthocenter,escenter)and Fermat Point?In the provincial high school mathematics contest preliminary contest and the national high school mathematics league plane geometry examination questions,the investigation of "five hearts" is very common?Five centers and Fermat are also included in the National High School Mathematics League Competition Outline,the importance of which can be seen?Taking the high school mathematics contest in Hubei Province as an example,we inspected the orthocenter in 2008,incenter in 2012,excenter in 2009 and 2017,and barycenter in 2013 and 2018?Combining with my problem solving and teaching practice,this paper mainly introduces nine common special points of triangle(five centers of triangle,Fermat point,Nagel point,Brocard point,regular point)and their properties,and shows the flexible application of the special point nature through abundant examples.By means of the drawing tool of special points of triangles in the geometric sketchpad,the vector representation of special points,coordinates of special points,distance between special points and vertices of triangles,formula of center distance between special points,distribution law of special points and geometric inequalities of special points are also studied?Using these properties,we get more methods to solve plane geometry problems,and summarize some typical questions in this paper.