Problem

Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 4

Tags: geometry, rectangle, incenter, ratio, geometry proposed



Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r+r_0=R.\] Soewono, Bandung