Geometry Problem 1114: Triangle, Perpendicular Bisector, 90 Degrees, Circle, Concyclic Points, Cyclic Quadrilateral. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a right triangle ABC with AA₁ and AA₂ trisectors of angle BAC, similarly CC₁ and CC₂ trisectors of angle ACB. AA₁ meets CC₂ and CC₁ at C₃ and B₁, respectively. AA₂ meets CC₂ and CC₁ at B₂ and A₃, respectively. B₁B₂ meets AC at B₃. Prove that angle A₃B₃C₃ = 90 degrees.

The figure below shows a right triangle ABC with AA₁ and AA₂ trisectors of angle BAC, similarly CC₁ and CC₂ trisectors of angle ACB. AA₁ meets CC₂ and CC₁ at C₃ and B₁, respectively. AA₂ meets CC₂ and CC₁ at B₂ and A₃, respectively. B₁B₂ meets AC at B₃. Prove that (1) angle A₃B₃C₃ = 90 degrees (2) .