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    OlympiadMediumInequalities10–12

    Geometric Inequality

    Desigualdade Geométrica

    In a triangle with sides a,b,ca, b, c and area SS, prove that S≤a⋅b2S \le \frac{a \cdot b}{2}.

    The area of a triangle is 12absin⁡C\tfrac{1}{2}ab \sin C, and sin⁡C≤1\sin C \le 1.

    Solution

    Step 1 of 5

    1. 1.S=12⋅a⋅b⋅sin⁡CS = \tfrac{1}{2} \cdot a \cdot b \cdot \sin C.