Question

17£®ÏÂÁÐÊÇÓйØÈý½ÇÐÎABCµÄ¼¸¸öÃüÌ⣬
¢ÙÈôtanA+tanB+tanC£¾0£¬Ôò¡÷ABCÊÇÈñ½ÇÈý½ÇÐΣ»
¢ÚÈôsin2A=sin2B£¬Ôò¡÷ABCÊǵÈÑüÈý½ÇÐΣ»
¢ÛÈô£¨$\overrightarrow{AB}$+$\overrightarrow{AC}$£©•$\overrightarrow{BC}$=0£¬Ôò¡÷ABCÊǵÈÑüÈý½ÇÐΣ»
¢ÜÈôcosA=sinB£¬Ôò¡÷ABCÊÇÖ±½ÇÈý½ÇÐΣ» 
ÆäÖÐÕýÈ·ÃüÌâµÄ¸öÊýÊÇ£¨¡¡¡¡£©

• A£® .1
• B£® .2
• C£® 3
• D£® 4

Answer 1

·ÖÎö ¢Ù¸ù¾ÝÁ½½ÇºÍ²îµÄÕýÇй«Ê½½øÐÐÅжϣ®¢Ú¸ù¾ÝÈý½Çº¯ÊýµÄ±¶½Ç¹«Ê½½øÐл¯¼òÅжϣ®¢Û¸ù¾ÝÏòÁ¿ÊýÁ¿»ýµÄÓ¦ÓÃÈ¥Åжϣ®¢Ü¸ù¾ÝÈý½Çº¯ÊýµÄÓÕµ¼¹«Ê½½øÐл¯¼òÅжϣ®

½â´ð ½â£º¢Ù¡ßtanA+tanB=tan£¨A+B£©£¨1-tanAtanB£©£¬
¡àtanA+tanB+tanC=tan£¨A+B£©£¨1-tanAtanB£©+tanC=tanAtanBtanC£¾0£¬
¡àA£¬B£¬CÊÇ¡÷ABCµÄÄڽǣ¬¹ÊÄڽǶ¼ÊÇÈñ½Ç£¬¹Ê¢ÙÕýÈ·£»
¢Ú¡ßsin2A=sin2B
¡àsin2A-sin2B=cos£¨A+B£©sin£¨A-B£©=0
¡àcos£¨A+B£©=0»òsin£¨A-B£©=0
¡àA+B=$\frac{¦Ð}{2}$»òA=B£¬
Èôsin2A=sin2B£¬Ôò¡÷ABCÊǵÈÑüÈý½ÇÐλòÊÇÖ±½ÇÈý½ÇÐΣ»¹Ê¢Ú´íÎó
¢ÛÈô£¨$\overrightarrow{AB}$+$\overrightarrow{AC}$£©•$\overrightarrow{BC}$=0£¬
Ôò£¨$\overrightarrow{AB}$+$\overrightarrow{AC}$£©•£¨$\overrightarrow{AC}$-$\overrightarrow{AB}$£©=0£¬
¼´|$\overrightarrow{AB}$|²-|$\overrightarrow{AC}$|²=0£¬
Ôò|$\overrightarrow{AB}$|²=|$\overrightarrow{AC}$|²£¬¼´|$\overrightarrow{AB}$|=|$\overrightarrow{AC}$|£¬
ÔòAB=AC£¬Ôò¡÷ABCÊǵÈÑüÈý½ÇÐΣ»ÕýÈ·£¬¹Ê¢ÛÕýÈ·£¬
¢ÜÈôcosA=sinB£¬ÔòsinB=cosA=sin£¨$\frac{¦Ð}{2}-A$£©£¬¡à$B=\frac{¦Ð}{2}-A»òB+\frac{¦Ð}{2}-A=¦Ð$£¬
¼´A+B=$\frac{¦Ð}{2}$»òB-A=$\frac{¦Ð}{2}$£¬Ôò¡÷ABC²»Ò»¶¨ÎªÖ±½ÇÈý½ÇÐΣ¬¹Ê¢Ü´íÎó£¬
¹ÊÑ¡£ºB

µãÆÀ ±¾ÌâÖ÷Òª¿¼²éÃüÌâµÄÕæ¼ÙÅжϣ¬Éæ¼°Èý½ÇÐÎÐÎ×´µÄÅжϣ¬ÀûÓÃÈý½Çº¯ÊýµÄÓÕµ¼¹«Ê½ºÍÈý½Ç¹«Ê½Êǽâ¾ö±¾ÌâµÄ¹Ø¼ü£®ÒªÇóÊìÁ·ÕÆÎÕÈý½Çº¯ÊýµÄÔËË㹫ʽ£¬¿¼²éѧÉúµÄÔËËãÄÜÁ¦£®