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projections of point on circumcirle on radii form equilateral of fixed area

Source: Czech-Polish-Slovak Junior Match 2012, Individual p3 CPSJ

March 11, 2020
circumcircleEquilateralfixedarea of a trianglegeometryprojections

Problem Statement

Different points A,B,C,DA, B, C, D lie on a circle with a center at the point OO at such way that ∠AOB\angle AOB =∠BOC== \angle BOC = ∠COD=\angle COD = 60o60^o. Point PP lies on the shorter arc BCBC of this circle. Points K,L,MK, L, M are projections of PP on lines AO,BO,COAO, BO, CO respectively . Show that (a) the triangle KLMKLM is equilateral, (b) the area of triangle KLMKLM does not depend on the choice of the position of point PP on the shorter arc BCBC
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