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AP/AD >= 1 - BC/(AB + CA), touchpoints with incircle

Source: Indonesia INAMO Shortlist 2009 G4 https://artofproblemsolving.com/community/c1101409_

December 10, 2021
geometric inequalitygeometryincircle

Problem Statement

Let D,E,FD, E, F, be the touchpoints of the incircle in triangle ABCABC with sides BC,CA,ABBC, CA, AB, respectively, . Also, let ADAD and EFEF intersect at PP. Prove that APAD≥1−BCAB+CA\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}.
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