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说明:本文引理及证明中出现的线段均为有向线段.如图1,直线l_1上有两定点A、D及动点P,直线l_2上有两定点B、C及动点Q满足(AP)/(PD)=(BQ)/(QC),并补充定义点p与D重合时,点Q与C重合.引理1给定实数u,若点R在PQ上使(PR)/(RQ)=u,则R的轨迹是直线.引理2设AQ与BP交于点S,特别地,当点P与A重合时补充点S的位置为P、Q分别向A、B运动时点S所趋于的极限,并设AB、CD的中点分别为M、N.则点S的轨迹为平行于MN的直线.
As shown in Figure 1, there are two fixed points A and D and a moving point P on the straight line l_1, and two fixed points B and C and a moving point Q on the straight line l_2 satisfy ( (PQ) / (PD) = (BQ) / (QC), and complementing the definition point p coincides with D, point Q coincides with C. Lemma 1 gives the real u, and if point R is on PQ such that (PR) / (RQ) = u, the trajectory of R is a straight line. Lemma 2 Let AQ and BP be at point S, in particular, when point P and A coincide, position of complementary point S is P and Q move to A and B, respectively When the point S tends to the limit, and set AB, CD, respectively, the midpoint of M, N. The point S trajectory parallel to the MN’s line.