Problem 349. Triangle, Cevian, Incircles, Tangents, Tangency Points.
Level: High School, College, SAT Prep.

The figure shows a triangle ABC with point D on side AC. Circles 1, 2, and 3 are the incircles of triangles ABC, ABD, and BDC, respectively. QS is the common tangent to circles 2 and 3. QS intersects to cevian BD at R. If E,G,H,F,P,N,M,Q, and S are points of tangency, prove that EG = FH = DP = DN = RM = QR.