Infographic Geometry Problem 1062: Scalene Triangle with an angle of 60 degrees, Euler Line, Orthocenter, Circumcenter, Distance, Congruence, Transversal. Level: High School, SAT Prep, College, Mathematics Education

The figure shows a scalene triangle ABC with the circumcircle of center O and angle B = 60 degrees. The Euler line OH intersects arc AB, AB, BC, and arc BC at C₂, C₁, A₁, and A₂, respectively. Prove that (1) OH = C₁C₂ - A₁A₂; (2) OH = AC₁ - CA₁; (3) OH = AB - BC; (4) OC₁ = HA₁.