Starting with triangle ABC, locate points D, E, and F on sides CB, AC, and BA, respectively such that the ratio CD:CB=AE:AC=BF:BA=1:3. Draw segments AD, BE, and CF to form triangle PQR (where P,Q, and R are intersection points of segments AD, BE, and CF).

Using Geometer's Sketchpad, explore relationships that exist among the various triangles within the construction. What relationships exist between PQR and ABC? Prove your conjectures.

Problem adapted from:
Jiang, Z. and McClintock, E. (1997). Using the Geometer's Sketchpad with preservice teachers. In J. King and D. Schattschneider (Eds.) Geometry turned on: Dynamic software in learning, teaching, and research. (MAA notes 41.) Washington, DC: Mathematical Association of America.