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Geometric inequality

Source: Iran TST 2012-First exam-2nd day-P6

April 24, 2012
inequalitiesgeometrygeometric transformationreflectionlimitanalytic geometrycomplex numbers

Problem Statement

The pentagon ABCDEABCDE is inscirbed in a circle ww. Suppose that wa,wb,wc,wd,wew_a,w_b,w_c,w_d,w_e are reflections of ww with respect to sides AB,BC,CD,DE,EAAB,BC,CD,DE,EA respectively. Let A′A' be the second intersection point of wa,wew_a,w_e and define B′,C′,D′,E′B',C',D',E' similarly. Prove that 2≤SA′B′C′D′E′SABCDE≤3,2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3, where SXS_X denotes the surface of figure XX.
Proposed by Morteza Saghafian, Ali khezeli
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