MathDB

2

Part of 2023 Turkey Olympic Revenge

Problems(1)

Conditional Geometry With Parallel Lines

Source: Turkey Olympic Revenge 2023 P2

3/6/2023
Let ABCABC be a triangle. A point DD lies on line BCBC and points E,FE,F are taken on AC,ABAC,AB such that DE∥ABDE \parallel AB and DF∥ACDF\parallel AC. Let G=(AEF)∩(ABC)≠AG = (AEF) \cap (ABC) \neq A and I=(DEF)∩BC≠DI = (DEF) \cap BC\neq D. Let HH and OO denote the orthocenter and the circumcenter of triangle DEFDEF. Prove that A,O,IA,O,I are collinear if and only if G,H,IG,H,I are collinear.
Proposed by Kaan Bilge
geometryconditional geometryTriangle Geometryolympic revenge