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Turkey NMO 2007 1st Round - P05 (Geometry)

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October 4, 2012
geometry

Problem Statement

Let CC and DD be points on the semicircle with center OO and diameter ABAB such that ABCDABCD is a convex quadrilateral. Let QQ be the intersection of the diagonals [AC][AC] and [BD][BD], and PP be the intersection of the lines tangent to the semicircle at CC and DD. If m(AQB^)=2m(COD^)m(\widehat{AQB})=2m(\widehat{COD}) and ∣AB∣=2|AB|=2, then what is ∣PO∣|PO|?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 1+32<spanclass=′latex−bold′>(D)</span> 1+322<spanclass=′latex−bold′>(E)</span> 233 <span class='latex-bold'>(A)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ \frac{1+\sqrt 3} 2 \qquad<span class='latex-bold'>(D)</span>\ \frac{1+\sqrt 3}{2\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ \frac{2\sqrt 3} 3
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