Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to AB.

Prove that Barney will eventually return to his starting point.

How many times will Barney reach a wall before returning to his starting point?

Explore and discuss for various starting points on line BC, including points

exteriorto segment BC.Discuss and prove any mathematical conjectures you find in the situation.

Will Barney always return to his starting point?

If so, will Barney always travel the same distance?

Does the path create a pattern of similar triangles?

Where might his starting point be so that the path would create

congruenttriangles?What if Barney started at a point inside the triangle? Anything special about the path if he starts at the centroid? the othocenter?

Extend the sides of the triangle and let Barney begin his journey from a point

outsidethe triangle. His path is slightly redefined -- rather than "bouncing" when he comes to a wall, he crosses the wall to change direction and continues to travel a path parallel to a side of the triangle. Construct a GSP image and explore.The problem is rather open-ended. Don't give up on it too early.

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