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P33 [Geometry] - Turkish NMO 1st Round - 2001

Source:

April 23, 2014
geometrycircumcircletrigonometrytrig identitiesLaw of Sines

Problem Statement

Let ABCABC be a triangle such that ∣AC∣=1|AC|=1 and ∣AB∣=2|AB|=\sqrt 2. Let MM be a point such that ∣MA∣=∣AB∣|MA|=|AB|, m(MAB^)=90∘m(\widehat{MAB}) = 90^\circ, and CC and MM are on the opposite sides of ABAB. Let NN be a point such that ∣NA∣=∣AX∣|NA|=|AX|, m(NAC^)=90∘m(\widehat{NAC}) = 90^\circ, and BB and NN are on the opposite sides of ACAC. If the line passing throung AA and the circumcenter of triangle MANMAN meets [BC][BC] at FF, what is ∣BF∣∣FC∣\dfrac {|BF|}{|FC|}?
<spanclass=′latex−bold′>(A)</span> 22<spanclass=′latex−bold′>(B)</span> 23<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 32 <span class='latex-bold'>(A)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 3\sqrt 2
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