Problem

Source: Iranian National Olympiad (3rd Round) 2007

Tags: geometry, geometric transformation, reflection, parallelogram, geometry proposed



Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.