Geometry · Incenter · Angle bisector theorem · Vectors · Barycentric coordinates · Cevians
Let \(ABC\) be a triangle and let \(a\), \(b\), \(c\) denote the lengths of the sides opposite the vertices \(A\), \(B\), \(C\) respectively. Prove that a point \(S\) is the centre of the inscribed circle of \(ABC\) if and only if
\[ a \cdot \overrightarrow{AS} + b \cdot \overrightarrow{BS} + c \cdot \overrightarrow{CS} = \vec{0} . \]
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Serbian Regional Competition (Okruzno takmicenje) 1998, high school grade I, problem 3. The 1998 regional paper was not split into categories. Organized by the Mathematical Society of Serbia (DMS). Source