打字猴:1.70101052e+09
1701010520 设,且O为ΔABC的重心,则,且,记为S,那么,
1701010521
1701010522
1701010523
1701010524
1701010525 即:
1701010526
1701010527
1701010528
1701010529
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1701010531
1701010532 同理可得,。
1701010533
1701010534
1701010535 所以,则:
1701010536
1701010537
1701010538
1701010539
1701010540 3.过重心截取三角形两边长,形成的比例的倒数和为定值
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1701010543 【问题】如图9-18所示,已知点O是ΔABC的重心,过O作直线与AB、AC两边分别交于M、N两点,且,则:
1701010544
1701010545
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1701010547
1701010548
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1701010550
1701010551 图9-18 三角形重心应用
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1701010553 【分析】
1701010554
1701010555 点O是ΔABC的重心,知:
1701010556
1701010557
1701010558
1701010559
1701010560
1701010561 得,从而有:
1701010562
1701010563
1701010564
1701010565
1701010566 又M、N、G三点共线(A不在直线AM上),于是存在λ和μ,使得:
1701010567
1701010568
1701010569
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