Cho bốn điểm A,B,C,D trong mặt phẳng. Chứng minh rằng
−−→AB.−−→CD+−−→BC.−−→AD+−−→CA.−−→BD=0.
Ta có: −−→AD=−−→AB+−−→BC+−−→CD
−−→CA=−−→BA−−−→BC
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−−→BD=−−→BC+−−→CD
Ta có:
−−→AB.−−→CD+−−→BC.−−→AD+−−→CA.−−→BD=−−→AB.−−→CD+−−→BC(−−→AB+−−→BC+−−→CD)+(−−→BA−−−→BC)(−−→BC+−−→CD)
=−−→AB.−−→CD+−−→BC.−−→AB+−−→BC2+−−→BC.−−→CD+−−→BA.−−→BC+−−→BA.−−→CD−−−→BC2−−−→BC.−−→CD
=(−−→AB.−−→CD+−−→BA.−−→CD)+(−−→BC.−−→AB+−−→BA.−−→BC)+(−−→BC2−−−→BC2)+(−−→BC.−−→CD−−−→BC.−−→CD)=0
⇒−−→AB.−−→CD+−−→BC.−−→AD+−−→CA.−−→BD=0 (đpcm)
