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P21 [Geometry] - Turkish NMO 1st Round - 2013

Source:

April 16, 2013
geometrycircumcircleincentercyclic quadrilateralangle bisector

Problem Statement

Let DD and EE be points on side [AB][AB] of a right triangle with m(C^)=90∘m(\widehat{C})=90^\circ such that ∣AD∣=∣AC∣|AD|=|AC| and ∣BE∣=∣BC∣|BE|=|BC|. Let FF be the second intersection point of the circumcircles of triangles AECAEC and BDCBDC. If ∣CF∣=2|CF|=2, what is ∣ED∣|ED|?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 1+2<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 22<spanclass=′latex−bold′>(E)</span> None of above <span class='latex-bold'>(A)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 1+\sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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