探求分成等积四边形的个数

• 作 者: 周士藩
• 机 构: 苏州大学数学科学学院,
• 出 处: 中学生数学
• 发表日期: 2014年14期
• 关键词: 题设 确定值 分点 可证

大家都知道一个定理:“等高三角形的面积之比等于底边之比”.本文目的是充分利用这一性质,探求等积四边形的个数.例1如图1的四边形ABCD,它的面积为S,且AE∶EB=CF∶FD=p∶q,则S四边形AECF=p p+qS 1S四边形EBFD=q p+q2图1图2分析与解在图1中,连结AF、AC及EC,得图2,据题设可知:因AB=AE+EB,CD=CF+FD,故AE∶AB=p∶(p+q),且CF∶CD=p∶(p+q),于是由定理得S△AEC∶S△ABC=p∶(p+q).故S△AEC=p p+qS△ABC,

We all know a theorem: “The ratio of the area of ​​the contour triangle is equal to the ratio of the base ”. The purpose of this paper is to make full use of this property, to explore the number of quadrilateral.Example 1 quadrilateral ABCD in Figure 1, it Of the area S, and AE: EB = CF: FD = p: q, then the S quadrilateral AECF = p p + qS 1S quadrilateral EBFD = q p + q2 Figure 1 Figure 2 Analysis and Solutions In Figure 1, AC and EC, Figure 2, according to the title set known: AB = AE + EB, CD = CF + FD, so AE: AB = p: (p + q), and CF: CD = p: (p + q ), So by the theorem S △ AEC: S △ ABC = p: (p + q). So S △ AEC = p p + qS △ ABC,