1997年全国高中数学联赛一试第四题为 :设双曲线 xy =1的两支 C1、C2 ,正三角形PQR的三个顶点位于此双曲线上 ,求证 :P、Q、R不能都在双曲线的同一支上 .本文先给出赛题的几种证法 ,然后给出等轴双曲线上的三点构成正三角形的一个充要条件 .图 ( 1 )证法 1　假设 P、Q、R在双曲 In the 1997 National High School Mathematical League Tournament, the fourth question was: suppose the two C1 and C2 hyperbolic xy = 1, and the three vertices of the positive triangle PQR are located on this hyperbola. Proof: P, Q, R cannot all be double On the same branch of the curve, this paper first gives several proofs of the title, and then gives a necessary and sufficient condition for the three points forming the equilateral triangle on the isometric hyperbola. (1) Proof 1 Assumption P, Q, R is in double