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题目如图1,已知锐角△ABC的外接圆为⊙O,AD为⊙O的直径,过点B、C且垂直于BC的直线与CA、BA的延长线分别交于点E、F.证明:∠ADF=∠BED.此题是第四届(2013)陈省身杯全国高中数学奥林匹克第5题,文[1]中提供了组委会给出的参考答案,利用的是位似变换法,笔者经探究再给出有别于组委会所提供的参考答案的两种新证法,供参考赏析.
The subject is shown in Fig.1. It is known that the circumcircle of the acute angle △ ABC is ⊙O, and the diameter of AD is ⊙O. The straight lines crossing point B, C and perpendicular to BC and the extension lines of CA and BA are respectively intersected with points E and F. Proof: ∠ADF = ∠BED.This question is the fourth (2013) Chen provincial body cup national high school mathematics Olympiad fifth question, the text [1] provides the reference given by the organizing committee, using the bit-like transformation , The author gives another two different proofs that are different from the reference answers provided by the Organizing Committee after exploration, for reference appreciation.