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Geometry Mathley 2.1 OM >= R , S_bS_c = S_aS_b+S_aS_c

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June 6, 2020
area of trianglegeometric inequalitygeometryEquilateralcircumcircleareas

Problem Statement

Let ABCABC be an equilateral triangle with circumcircle of center OO and radius RR. Point MM is exterior to the triangle such that SbSc=SaSb+SaScS_bS_c = S_aS_b+S_aS_c, where Sa,Sb,ScS_a, S_b, S_c are the areas of triangles MBC,MCA,MABMBC,MCA,MAB respectively. Prove that OM≥ROM \ge R.
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