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Question Number 140735 by ZiYangLee last updated on 12/May/21
Usingvectormethod,provethatthreemedianofatriangleareconcurrent.
Answered by MJS_new last updated on 12/May/21
letA=(00)∧B=(c0)∧C=(ph)Ma=B+C2=(c+p2h2)Mb=A+C2=(p2h2)Mc=A+B2=(c20)va=C−B=(p−ch)vb=A−C=(−p−h)vc=B−A=(c0)na=(−hp−c)nb=(h−p)nc=(0c)mediansma:X=Ma+λana⇔y=c−phx−c2−h2−p22h(I)mb:X=Mb+λbnb⇔y=−phx+h2+p22h(II)mc:X=Mc+λcnc⇔x=c2insertingx=c2inbothequationsIandIIwegetthesamevalueforyy=−cp+h2+p22h⇒ma∩mb∩mc=(c2−cp+h2+p22h)
Answered by peter frank last updated on 12/May/21
Commented by peter frank last updated on 12/May/21
Themedianoftrianglecutsanothermedianinratio(2:1)fromAD→E(B→+C→2)P=1×A+2(B→+C→)23=A→+B→+C→3fromBE→(A→+C→2)P=1×B+2(A→+C→)23=A→+B→+C→3fromFC→(A→+B→2)P=1×C+2(A→+B→)23=A→+B→+C→3henceitsconcurent
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