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Turkey NMO 2007 1st Round - P17 (Geometry)

Source:

October 4, 2012
geometrycircumcircle

Problem Statement

Let KK be the point of intersection of ABAB and the line touching the circumcircle of △ABC\triangle ABC at CC where m(A^)>m(B^)m(\widehat {A}) > m(\widehat {B}). Let LL be a point on [BC][BC] such that m(ALB^)=m(CAK^)m(\widehat{ALB})=m(\widehat{CAK}), 5∣LC∣=4∣BL∣5|LC|=4|BL|, and ∣KC∣=12|KC|=12. What is ∣AK∣|AK|?
<spanclass=′latex−bold′>(A)</span> 42<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 9<spanclass=′latex−bold′>(E)</span> None of the above <span class='latex-bold'>(A)</span>\ 4\sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 9 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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