N.T.TUAN wrote: Let $M$ be a point inside a triangle $ABC$ . The lines $AM , BM , CM$ intersect $BC, CA, AB$ at $A_{1}, B_{1}, C_{1}$, respectively. Assume that $S_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B}$ . Prove that one of the lines $AA_{1}, BB_{1}, CC_{1}$ is a median of the triangle $ABC.$ See http://www.mathlinks.ro/Forum/viewtopic.php?t=6074 (in the very beginning of my solution, I show that the problem is equivalent to yours). Darij