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.
Geometry problem solving
Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."