IMO 2018 Shortlist G7
Let O be the ir um entre, and Ω be the ir um ir le of an a ute-angled triangle ABC. Let P be an arbitrary point on Ω, di...
Category: Geometry
Problem
Let O be the ir um entre, and Ω be the ir um ir le of an a ute-angled triangle ABC. Let P be an arbitrary point on Ω, distin t from A, B, C, and their antipodes in Ω. Denote the ir um entres of the triangles AOP, BOP, and COP by OA, OB, and OC, respe tively. The lines ℓA, ℓB, and ℓC perpendi ular to BC, CA, and AB pass through OA, OB, and OC, respe tively. Prove that the ir um ir le of the triangle formed by ℓA, ℓB, and ℓC is tangent to the line OP. (Russia) Shortlisted problems 7