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inradii inequality

Source: Yugoslav TST 1998 P2

May 15, 2021
inequalitiesgeometric inequalityGeometric Inequalitiesgeometry

Problem Statement

In a convex quadrilateral ABCDABCD, the diagonal ACAC intersects the diagonal BDBD at its midpoint SS. The radii of incircles of triangles ABS,BCS,CDS,DASABS,BCS,CDS,DAS are r1,r2,r3,r4r_1,r_2,r_3,r_4, respectively. Prove that ∣r1−r2+r3−r4∣≤18∣AB−BC+CD−DA∣.|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.
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