论文部分内容阅读
一九八六年全国初中数学竞赛题第三题: “设P、Q为线段BC上两定点,且BP=CQ,A为BC外一动点(如图)。当点A运动到使∠BAP=∠CAQ时,△ABC是什么三角形?试证明你的结论”。这是一道源于教材、高于教材、难度适中、证法灵活、既考基础、又考能力的不可多得的好题;也是一道较好的综合训练的范例。本刊编辑部仅在十天之内就先后收到不少本题证法的来稿。现根据湖北洪湖县侯书清、湖南常德地区刘茂林、安徽宿州陈新昌、辽宁锦州张士贞、贵州普安石又栋、广西百色地区叶添蕃、湖北钟祥县贾双喜等同志的来稿综合成如下12种证法,供同行参考。首先不难猜想其为等腰三角形(此题实际上是由《几何》第一册P_(119)习题10演变而来),再看其证明:
In the 1986 National Junior High School Mathematics Competition Question 3: “Set P and Q to two fixed points on the line BC, and BP=CQ, A is a point outside BC (as shown in the figure). When point A moves to make BAP = ∠ CAQ, what triangle is △ABC? Try to prove your conclusion." This is a rare good question that stems from the teaching materials, higher than the teaching materials, moderate difficulty, flexible evidence, both the examination of the basics, and the examination ability; it is also a good example of comprehensive training. The editorial department of this periodical has received many submissions of this authentication method within 10 days. Based on the contributions of Hou Shuqing of Hubei Honghu County, Liu Maolin of Changde District of Hunan Province, Chen Xinchang of Changde District of Hunan Province, Zhang Shizhen of Jinzhou of Liaoning Province, Shi Youdong of Guizhou Province, Ye Tianfan of Baise Prefecture of Guangxi Province, and Jia Shuangxi of Zhongxiang County of Hubei Province, the following 12 certification methods are available for use by peers. reference. First of all, it is not difficult to guess that it is an isosceles triangle (this question is actually evolved from the geometry of the first volume of Exercise P_ (119) Exercise 10), and then see its proof: