Barney is in the triangular room shown here. He walks from a point on AC parallel to BC. When he reaches AB, he turns and walks parallel to AC. When he reaches BC, he turns and walks parallel to AB. How many times will Barney reach a wall before returning to his starting point?
Open GSP and construct a triangle. Pick a point on one side of the triangle. Click on the opposite side and construct a parallel line. Click on the point that touches the side of the triangle and the opposite side. Construct another parallel line. Continue this process until the line segments return to the original point X. Experiment with moving the triangle around and observe what happens to the points on the line
Because the line segments that Barney follows are parallel to opposite sides, similar triangles are formed within the triangular room. Since they have the same measure angles, the path that Barney takes always follows the same angle of the original triangle. Barney will always return to the same point at which he began because of this fact.
Try beginning the point outside the triangle and notice what happens. Again, because these triangles are similaer (same measure angles), the line segments that are formed with follow the same path and result in a return to his original position.