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one of the lines $AA_1 , BB_1, CC_1$ is a median of ABC

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

March 30, 2007
geometry proposedgeometry

Problem Statement

Let MM be a point inside a triangle ABCABC . The lines AM,BM,CMAM , BM , CM intersect BC,CA,ABBC, CA, AB at A1,B1,C1A_{1}, B_{1}, C_{1}, respectively. Assume that SMAC1+SMBA1+SMCB1=SMA1C+SMB1A+SMC1BS_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B} . Prove that one of the lines AA1,BB1,CC1AA_{1}, BB_{1}, CC_{1} is a median of the triangle ABC.ABC.
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