Problem

Source: Iranian National Olympiad (3rd Round) 2004

Tags: geometry, circumcircle, inradius, incenter, ratio, trigonometry, IMO Shortlist



Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB+AC=3BC$.