论文部分内容阅读
本文约定:若凸n边形的n边(或延长线)均与圆锥曲线相切,则称此凸n边形为圆锥曲线的外切凸n边形.笔者最近探究发现圆锥曲线外切凸n边形的一个性质,现将结果陈述如下,供大家参考.命题1若三角形△A_1A_2A_3的三边A_1A_2、A_2A_3、A_3A_1(或其延长线),与圆锥曲线Γ分别相切于点T_1、T_2、T_3,则A_1T_1/T_1A_2·A_2T_2/T_2A_2·A_3T_3/T_3A_1=1.证明:(1)当Γ为椭圆时,如图1,设其标准方程为x~2/a~2+y~2/b~2=1(a>b>0),T_i(acosθ_i,nsinθ_i),其中θ_i-θ_i≠kπ,(i≠j,i,j=1,2,3),
This paper stipulates that if the n-side (or extension line) of a convex n-edge is tangent to the conic curve, this convex n-edge is called the conic-convex n-gon of the conic curve. A property of the n-sided shape, the results are presented below for your reference. Proposition 1 If the three sides A_1A_2, A_2A_3, A_3A_1 (or their extension lines) of the triangle ΔA_1A_2A_3, and the conic curve Γ are tangent to the points T_1, T_2, respectively T_3, then A_1T_1/T_1A_2A_2T_2/T_2A_2A_3T_3/T_3A_1=1. Prove that: (1) When Γ is an ellipse, as shown in Figure 1, set its standard equation as x~2/a~2+y~2/ b~2=1 (a>b>0), T_i(acosθ_i, nsinθ_i), where θ_i-θ_i≠kπ, (i≠j,i,j=1,2,3),