Geometry · Vectors · Centroid · Parallelogram · Midpoints
Let \(X\) be a point in the interior of triangle \(ABC\), and let \(T\) be the centroid of that triangle. Points \(M\) and \(N\) lie on side \(BC\), points \(P\) and \(Q\) lie on side \(CA\), and points \(R\) and \(S\) lie on side \(AB\), in such a way that
\[ MQ \parallel AB, \qquad PS \parallel BC, \qquad RN \parallel CA, \qquad MQ \cap PS \cap RN = \{X\}. \]
Let \(A_1\), \(B_1\), \(C_1\) be the midpoints of the segments \(MN\), \(PQ\), \(RS\) respectively. Prove that
\[ \overrightarrow{XA_1} + \overrightarrow{XB_1} + \overrightarrow{XC_1} = \frac{3}{2}\,\overrightarrow{XT}. \]
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Serbian Municipal Competition 2009, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source