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Inequalities in convex quadrilaterals

Source: Sharygin First Round 2013, Problem 11

April 8, 2013
inequalitiesgeometrygeometry proposed

Problem Statement

a) Let ABCDABCD be a convex quadrilateral and r1≤r2≤r3≤r4r_1 \le r_2 \le r_3 \le r_4 be the radii of the incircles of triangles ABC,BCD,CDA,DABABC, BCD, CDA, DAB. Can the inequality r4>2r3r_4 > 2r_3 hold?
b) The diagonals of a convex quadrilateral ABCDABCD meet in point EE. Let r1≤r2≤r3≤r4r_1 \le r_2 \le r_3 \le r_4 be the radii of the incircles of triangles ABE,BCE,CDE,DAEABE, BCE, CDE, DAE. Can the inequality r2>2r1r_2 > 2r_1 hold?
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