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P05 [Geometry] - Turkish NMO 1st Round - 2005

Source:

October 26, 2013
geometrycircumcircleparallelogramperpendicular bisector

Problem Statement

Let MM be the intersection of diagonals of the convex quadrilateral ABCDABCD, where m(AMB^)=60∘m(\widehat{AMB})=60^\circ. Let the points O1O_1, O2O_2, O3O_3, O4O_4 be the circumcenters of the triangles ABMABM, BCMBCM, CDMCDM, DAMDAM, respectively. What is Area(ABCD)/Area(O1O2O3O4)Area(ABCD)/Area(O_1O_2O_3O_4)?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 32<spanclass=′latex−bold′>(D)</span> 1+232<spanclass=′latex−bold′>(E)</span> 1+32 <span class='latex-bold'>(A)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3}2 \qquad<span class='latex-bold'>(D)</span>\ \dfrac {1+2\sqrt 3}2 \qquad<span class='latex-bold'>(E)</span>\ \dfrac {1+\sqrt 3}2
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