2001年TI杯全国初中数学竞赛第14题是一道精彩的几何题,尤其是结论中三条线段成调和数列(即倒数成等差数列),倍受竞赛选手和老师的关注,给出多种证法,但都是纯几何证法.我们拟给出一种解析证法,并把它推广到圆锥曲线中去.1竞赛题的解析证法原题是:如图1,已知点P是⊙O外一点,PS The 14th issue of the 2001 TI Cup National Junior High School Mathematical Contest was a brilliant geometrical question, especially the conclusion that the three line segments became harmonic sequences (that is, the reciprocal number was a series of arithmetic progressions), which was greatly concerned by contestants and teachers and gave various certificates. Laws, but all are pure geometric proofs. We propose to give an analytical proof method and generalize it to the conic section. 1 The analytical proof of the contest title is: Figure 1. Known point P is ⊙O, PS