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P17 [Geometry] - Turkish NMO 1st Round - 2002

Source:

August 8, 2014
geometrytrapezoidincentersymmetry

Problem Statement

Let ABCDABCD be a trapezoid and a tangential quadrilateral such that AD∣∣BCAD || BC and ∣AB∣=∣CD∣|AB|=|CD|. The incircle touches [CD][CD] at NN. [AN][AN] and [BN][BN] meet the incircle again at KK and LL, respectively. What is ∣AN∣∣AK∣+∣BN∣∣BL∣\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}?
<spanclass=′latex−bold′>(A)</span> 8<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 16 <span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 9 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 16
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