Geometry · Centroid · Vectors in the plane · Ratios on the sides of a triangle · Affine combinations · Linear independence
On the sides \(BC\), \(CA\) and \(AB\) of a triangle \(ABC\) points \(A_1\), \(B_1\) and \(C_1\) are marked, respectively. Let \(T\) be the centroid of the triangle \(ABC\) and \(T_1\) the centroid of the triangle \(A_1B_1C_1\). Prove that \(T = T_1\) holds if and only if
\[ AC_1 : C_1B = BA_1 : A_1C = CB_1 : B_1A . \]
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Serbian Republic Competition (Republicko takmicenje) 2003, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source