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The area does not depend on the position of point

Source: International Zhautykov Olympiad 2012 - D1 - P1

January 31, 2012
geometrygeometry questionPlane Geometry

Problem Statement

An acute triangle ABCABC is given. Let DD be an arbitrary inner point of the side ABAB. Let MM and NN be the feet of the perpendiculars from DD to BCBC and ACAC, respectively. Let H1H_1 and H2H_2 be the orthocentres of triangles MNCMNC and MNDMND, respectively. Prove that the area of the quadrilateral AH1BH2AH_1BH_2 does not depend on the position of DD on ABAB.
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