MathDB
Nice Geometric inequality on angles sines and side lengths

Source: Iran Third Round MO 1997, Exam 1, P2

February 17, 2006
inequalitiestrigonometrygeometrycircumcirclegeometry proposed

Problem Statement

Show that for any arbitrary triangle ABCABC, we have sin⁡(A2)⋅sin⁡(B2)⋅sin⁡(C2)≤abc(a+b)(b+c)(c+a).\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.
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