Geometry · Area decomposition · Triangle inequalities · Circumradius · Chords of a circle
Let \(a\), \(b\), \(c\) be the side lengths of a triangle \(ABC\), let \(S\) be its area and \(R\) the radius of its circumscribed circle, and let \(M\) be a point in the interior of the triangle. Write \(d_a\), \(d_b\), \(d_c\) for the distances from \(M\) to the lines containing the sides \(BC\), \(CA\), \(AB\) respectively. Prove that
\[ 2S \cdot \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{R} \right) > d_a + d_b + d_c . \]
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Serbian National Competition (Drzavno takmicenje) 2014, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source