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P29 [Geometry] - Turkish NMO 1st Round - 2013

Source:

April 16, 2013
geometrycircumcirclegeometric transformationreflection

Problem Statement

Let OO be the circumcenter of triangle ABCABC with ∣AB∣=5|AB|=5, ∣BC∣=6|BC|=6, ∣AC∣=7|AC|=7. Let A1A_1, B1B_1, C1C_1 be the reflections of OO over the lines BCBC, ACAC, ABAB, respectively. What is the distance between AA and the circumcenter of triangle A1B1C1A_1B_1C_1?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 29<spanclass=′latex−bold′>(C)</span> 1926<spanclass=′latex−bold′>(D)</span> 3546<spanclass=′latex−bold′>(E)</span> 353 <span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ \sqrt {29} \qquad<span class='latex-bold'>(C)</span>\ \dfrac {19}{2\sqrt 6} \qquad<span class='latex-bold'>(D)</span>\ \dfrac {35}{4\sqrt 6} \qquad<span class='latex-bold'>(E)</span>\ \sqrt {\dfrac {35}3}
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