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P33 [Geometry] - Turkish NMO 1st Round - 2004

Source:

November 12, 2013
geometrytrapezoidtrigonometrycircumcircleangle bisectorperpendicular bisector

Problem Statement

Let ABCDABCD be a trapezoid such that ∣AB∣=9|AB|=9, ∣CD∣=5|CD|=5 and BC∥ADBC\parallel AD. Let the internal angle bisector of angle DD meet the internal angle bisectors of angles AA and CC at MM and NN, respectively. Let the internal angle bisector of angle BB meet the internal angle bisectors of angles AA and CC at LL and KK, respectively. If KK is on [AD][AD] and ∣LM∣∣KN∣=37\dfrac{|LM|}{|KN|} = \dfrac 37, what is ∣MN∣∣KL∣\dfrac{|MN|}{|KL|}?
<spanclass=′latex−bold′>(A)</span> 6263<spanclass=′latex−bold′>(B)</span> 2735<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 521<spanclass=′latex−bold′>(E)</span> 2463 <span class='latex-bold'>(A)</span>\ \dfrac{62}{63} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{27}{35} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{2}{3} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{5}{21} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{24}{63}
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