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Turkey NMO 2008 1st Round - P17 (Geometry)

Source:

August 25, 2012
geometrygeometric transformationreflectionPythagorean Theorem

Problem Statement

Let the vertices AA and CC of a right triangle ABCABC be on the arc with center BB and radius 2020. A semicircle with diameter [AB][AB] is drawn to the inner region of the arc. The tangent from CC to the semicircle touches the semicircle at DD other than BB. Let CDCD intersect the arc at FF. What is FD|FD|?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 52<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \frac 52 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5