Online Geometry Problem 935: Infographic, Triangle, Cevian, Equal Incircles, Inscribed Circles. Level: High School, SAT Prep, College, Mathematics Education

The infographic below shows a triangle ABC with cevians BD and BE. If the incircles O₁, O₂, and O₃ of the triangles ABD, DBE, and EBC, respectively, are equal, prove that the incircles Q₁ and Q₂ of the triangles ABE and DBC are equal.
A cevian is a line segment with one endpoint on a vertex of a triangle and the other endpoint on the opposite side.