Question

⊙O为△ABC的外接圆，请仅用无刻度的直尺，根据下列条件分别在图1，图2中画出一条弦，使这条弦将△ABC分成面积相等的两部分（保留作图痕迹，不写作法）．                                         

（1）如图1，AC=BC；                                                                           

（2）如图2，直线l与⊙O相切于点P，且l∥BC．                                        

                                               

　                                                                                                      

Answer 1

【考点】作图—复杂作图；三角形的外接圆与外心；切线的性质．                 

【专题】作图题．                                                                              

【分析】（1）过点C作直径CD，由于AC=BC， =，根据垂径定理的推理得CD垂直平分AB，所以CD将△ABC分成面积相等的两部分；                                                                       

（2）连结PO并延长交BC于E，过点A、E作弦AD，由于直线l与⊙O相切于点P，根据切线的性质得OP⊥l，而l∥BC，则PE⊥BC，根据垂径定理得BE=CE，所以弦AE将△ABC分成面积相等的两部分．                   

【解答】解：（1）如图1，                                                                    

直径CD为所求；                                                                              

（2）如图2，                                                                                    

弦AD为所求．                                                                                  

                                               

【点评】本题考查了复杂作图：复杂作图是在五种基本作图的基础上进行作图，一般是结合了几何图形的性质和基本作图方法．解决此类题目的关键是熟悉基本几何图形的性质，结合几何图形的基本性质把复杂作图拆解成基本作图，逐步操作．也考查了切线的性质．