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(CE-EA)/\sqrt{AB}+(AF-FB)/\sqrt{BC}+(BD-DC)/\sqrt{CA} >= ..., incircle's touches

Source: Indonesia INAMO Shortlist 2009 G12 https://artofproblemsolving.com/community/c1101409_

December 10, 2021
geometrygeometric inequalityincircle

Problem Statement

In triangle ABCABC, the incircle is tangent to BCBC at DD, to ACAC at EE, and to ABAB at FF. Prove that: CE−EAAB+AF−FBBC+BD−DCCA≥BD−DCAB+CE−EABC+AF−FBCA\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}} +\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}
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