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利用三角形的面积关系可以简捷地证明几何题,下面举出例题。例1 如图1设P为∠AOB平分线上任一点,延AP,BP分别交BO,AO于D、C。求证1/AO+1/DO=1/BO+1/OC。证设1/2∠AOB=α。∵ S_(△ADO)=S_(△APO)+S_(△PDO), ∴ AO·ODsin2α=(AO+DO)·OPsinα。 2AO·DOcosα=(AO+DO)·OP。同理 2BO·COcosα=(BO+CO)·OP。于是 AO·DO/(BO·CO)=(AO+DO)/(BO+CO), ∴1/AO+1/DO=1/BO+1/CO。例2 如图2,H是锐角三角形ABC的垂心,求证BC/AH+CA/BH
Using the area relationship of the triangle can simply prove the geometry problem. The following examples are given. Example 1 Let P be any point on the AOB bisector line as shown in Figure 1. The extension AP and BP are delivered to BO and AO to D and C, respectively. Confirm 1/AO+1/DO=1/BO+1/OC. Set 1/2 ∠AOB=α. ∵ S_(ΔADO)=S_(ΔAPO)+S_(ΔPDO), ∴AO·ODsin2α=(AO+DO)·OPsinα. 2AO·DOcosα=(AO+DO)·OP. Similarly, 2BO·COcosα=(BO+CO)·OP. Then AO·DO/(BO·CO)=(AO+DO)/(BO+CO), ∴1/AO+1/DO=1/BO+1/CO. Example 2 As shown in Figure 2, H is the angulation of the acute-angled triangle ABC to verify BC/AH+CA/BH.