一些共点线问题的新证法

• 作 者: 侯明辉
• 出 处: 数学大世界(初中版)
• 发表日期: 2002年12期
• 关键词: 共点线 证法 三线共点 连结线 角平分线 定理证明 三条 托勒密定理 三式 女口

∵BD>0,∴CC′=0,从而C、C′两点重合,于是AD、BE、CF三线共点。有趣的是,应用这个定理可以得到一些共点线问题的新证法(所谓共点线就是指一些直线通过同一点)。例1 如图3,设AD、BE、CF为△ABC的三条中线,求证:AD、BE、CF三线共点。证明作△ABC的外接圆,延长AD、BE、CF分别交圆于G、H、M,并作如图连结线,则易知△AEH∽△BEC,△CEH∽△BEC,从而

∵BD>0, ∴CC′=0, so that C, C′ overlap, so the AD, BE, and CF triple points are common. It is interesting to use this theorem to get new proofs for some common-line problems (the so-called common-dot lines mean that some straight lines pass through the same point). Example 1 As shown in Figure 3, set the AD, BE, and CF to the three midlines of △ABC. Verify that the three points of AD, BE, and CF are common. Prove that △ ABC circumcircle, extend AD, BE, CF respectively circled in G, H, M, and for the connection line as shown, it is easy to know △ AEH ∽ △ △ BEC, △ CEH ∽ △ BEC, thus