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Part of 2006 National Olympiad First Round

Problems(1)

Turkey NMO 2006 1st Round - P01 (Geometry)

Source:

2/2/2013
Let ABCABC be an equilateral triangle. DD and EE are midpoints of [AB][AB] and [AC][AC]. The ray [DE[DE cuts the circumcircle of △ABC\triangle ABC at FF. What is ∣DE∣∣DF∣\frac {|DE|}{|DF|}?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 33<spanclass=′latex−bold′>(C)</span> 23(3−1)<spanclass=′latex−bold′>(D)</span> 23<spanclass=′latex−bold′>(E)</span> 5−12 <span class='latex-bold'>(A)</span>\ \frac 12 \qquad<span class='latex-bold'>(B)</span>\ \frac {\sqrt 3}3 \qquad<span class='latex-bold'>(C)</span>\ \frac 23(\sqrt 3 - 1) \qquad<span class='latex-bold'>(D)</span>\ \frac 23 \qquad<span class='latex-bold'>(E)</span>\ \frac {\sqrt 5 - 1}2
geometrycircumcirclesymmetrypower of a point