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P33 [Geometry] - Turkish NMO 1st Round - 2005

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October 26, 2013
geometrytrigonometrycyclic quadrilateral

Problem Statement

Let KK be the intersection of diagonals of cyclic quadrilateral ABCDABCD, where ∣AB∣=∣BC∣|AB|=|BC|, ∣BK∣=b|BK|=b, and ∣DK∣=d|DK|=d. What is ∣AB∣|AB|?
<spanclass=′latex−bold′>(A)</span> d2+bd<spanclass=′latex−bold′>(B)</span> b2+bd<spanclass=′latex−bold′>(C)</span> 2bd<spanclass=′latex−bold′>(D)</span> 2(b2+d2−bd)<spanclass=′latex−bold′>(E)</span> bd <span class='latex-bold'>(A)</span>\ \sqrt{d^2 + bd} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{b^2+bd} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{2bd} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{2(b^2+d^2-bd)} \qquad<span class='latex-bold'>(E)</span>\ \sqrt{bd}
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