Geometry Problem 1490: Triangle, Cevian, Incircle, Tangent, Measurement

The figure shows a triangle ABC and a cevian BD. O₁, O₂, and O₃ are the incircles of triangles ABD, ABC, and BDC. If BE = 4, and BG = 5, find the measure of DF.

See solution below

Incircle

Incircle is a circle within a triangle, that is tangent to each side.

Cevian

A cevian is any segment drawn from a vertex of a triangle to the opposite side or its extension.