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证明a=2b型(或a=1/2b型)命题是平面几何中较常见的一类证明题,证法繁多,涉及定理广泛,但众多的证法通常可分别归属于四条思路,掌握这种思路后,再证明此类命题,便会得心应手,挥洒自如。例如重心定理的证明便可由此找出至少16种证法,下面进行逐一介绍。命题:求证三角形重心与顶点的距离等于它与对边中点的距离的两倍。已知:△ABC的三条中线AD、BE、CF相交于点O,求证:AO=2OD(BO=20E、CO=20F) 思路一利用折半法就是把长线段(AO)二等分,再证明其中一份和短线段(OD)相等。证明时,取AO的中点P,证AP=OD或OP。=OD即可,证法如下:
It is proved that the a=2b type (or a=1/2b type) proposition is a more common type of proving problem in plane geometry. There are many proof methods and the theorem is extensive, but many proof methods can usually be attributed to four ideas. After this kind of train of thought, proving these propositions will be handy and swaying. For example, the proof of the center of gravity theorem can be used to find at least 16 kinds of proofs, the following one by one. Proposition: Verify that the distance between the center of gravity of the triangle and the vertex is equal to twice its distance from the midpoint of the opposite side. It is known that the three midline AD, BE, and CF of △ABC intersect at the point O. Proof: AO=2OD (BO=20E, CO=20F) The idea of using the halving method is to divide the long segment (AO) into two equal parts and prove it again. One of them is equal to the short-term segment (OD). When proof, take the midpoint P of the AO, and verify that AP = OD or OP. =OD, certification method is as follows: