2013 Rusanovsky Lyc. Olympiad, https://artofproblemsolving.com/community/c4t48f4h2607229_collinear_wanted_tangential_pentagon__2013_rusanovsky_lyceum_olympiad_78

 Let ABC be a triangle with <B=2<C, and I - its incenter. Prove that BI=AC-AB

Proof
Let BD be the angle bisector of B, D onto AC, E reflection of B about AI; it is onto AC and <AEI=<ABI=<ACB, so IE||BC, BCEI is an isosceles trapezoid, BI=CI=AC-AE=AC-AB, done.