Problem

Source: Iranian National Olympiad (3rd Round) 2008

Tags: geometry, circumcircle, geometric transformation, reflection, incenter, cyclic quadrilateral, geometry proposed



Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' = BB' = CC' = x$. Prove that: a) If $ ABC\sim A'B'C'$ then $ x = 2r$ b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x = R + d$ or $ x = R - d$. (In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d = OI$)