Geometry · Medians · Centroid · Area inequality · Equality case · Parallelogram
Let \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that
\[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \]
and determine for which triangles equality holds.
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Serbian Regional Competition (Okruzno takmicenje) 2003, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source