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The reflections of diagonal in the bissector

Source: Sharygin contest 2008. The correspondence round. Problem 9

September 3, 2008
geometrygeometric transformationreflectioncircumcircletrigonometryprojective geometrycyclic quadrilateral

Problem Statement

(A.Zaslavsky, 9--10) The reflections of diagonal BD BD of a quadrilateral ABCD ABCD in the bisectors of angles B B and D D pass through the midpoint of diagonal AC AC. Prove that the reflections of diagonal AC AC in the bisectors of angles A A and C C pass through the midpoint of diagonal BD BD (There was an error in published condition of this problem).
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