Question

⊙O為△ABC的外接圓，請(qǐng)僅用無(wú)刻度的直尺，根據(jù)下列條件分別在圖1，圖2中畫(huà)出一條弦，使這條弦將△ABC分成面積相等的兩部分（保留作圖痕跡，不寫(xiě)作法）．                                         

（1）如圖1，AC=BC；                                                                           

（2）如圖2，直線l與⊙O相切于點(diǎn)P，且l∥BC．                                        

                                               

　                                                                                                      

Answer 1

【考點(diǎn)】作圖—復(fù)雜作圖；三角形的外接圓與外心；切線的性質(zhì)．                 

【專(zhuān)題】作圖題．                                                                              

【分析】（1）過(guò)點(diǎn)C作直徑CD，由于AC=BC， =，根據(jù)垂徑定理的推理得CD垂直平分AB，所以CD將△ABC分成面積相等的兩部分；                                                                       

（2）連結(jié)PO并延長(zhǎng)交BC于E，過(guò)點(diǎn)A、E作弦AD，由于直線l與⊙O相切于點(diǎn)P，根據(jù)切線的性質(zhì)得OP⊥l，而l∥BC，則PE⊥BC，根據(jù)垂徑定理得BE=CE，所以弦AE將△ABC分成面積相等的兩部分．                   

【解答】解：（1）如圖1，                                                                    

直徑CD為所求；                                                                              

（2）如圖2，                                                                                    

弦AD為所求．                                                                                  

                                               

【點(diǎn)評(píng)】本題考查了復(fù)雜作圖：復(fù)雜作圖是在五種基本作圖的基礎(chǔ)上進(jìn)行作圖，一般是結(jié)合了幾何圖形的性質(zhì)和基本作圖方法．解決此類(lèi)題目的關(guān)鍵是熟悉基本幾何圖形的性質(zhì)，結(jié)合幾何圖形的基本性質(zhì)把復(fù)雜作圖拆解成基本作圖，逐步操作．也考查了切線的性質(zhì)．