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Turkey NMO 2006 1st Round - P29 (Geometry)

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February 3, 2013
geometrytrigonometry

Problem Statement

Let II be the center of incircle of △ABC\triangle ABC, and JJ be the center of excircle tangent to [BC][BC]. If m(B^)=45∘m(\widehat B) = 45^\circ, m(A^)=120∘m(\widehat A) = 120^\circ, and ∣IJ∣=3|IJ|=\sqrt 3, then what is ∣BC∣|BC|?
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 34<spanclass=′latex−bold′>(D)</span> 62<spanclass=′latex−bold′>(E)</span> 3−1 <span class='latex-bold'>(A)</span>\ \frac 32 \qquad<span class='latex-bold'>(B)</span>\ \frac {\sqrt 3}2 \qquad<span class='latex-bold'>(C)</span>\ \frac 34 \qquad<span class='latex-bold'>(D)</span>\ \frac {\sqrt 6}2 \qquad<span class='latex-bold'>(E)</span>\ \sqrt3 - 1
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