Online Geometry Problem 929: Triangle, Orthocenter, Circumcircle, Reflection, Concurrency. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a triangle ABC. A line that passes through the orthocenter H intersects AB, BC, and CA extended at D, E, and F, respectively. A₁ is the reflection of D on BC, B₁ is the reflection of E on AC, and C₁ is the reflection of E on AB. Prove that (1) A₁E, B₁F, and C₁D are concurrent at P. (2) P is on the circumcircle O.