Barney is in the triangular room with vertices A, B, and C. He walks from a point on segment BC parallel to segment AC. When he reaches segment AB, he turns and walks parallel to BC. When he reaches segment 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 exterior to segment BC. Discuss and prove any mathematical conjectures you find in the situation.
Let explore Barney's adventure according to the directions above.



Since Barney walks on a path that parallel to each segment, he will always return to the starting point. Clearly, we can see that Barney touches the segments and exterior of the triangle five times before returning to his original position. From the pictorial representation of his starting point on segment on BC, it looks as though his path has similar triangle at each vertex.
There are several way to prove that this conjecture holds true. We will explore one way using the interior of the triangle ABC while comparing it to the segments of the triangle. Look at the first triangle above closely. Do you see three parallelograms? Since Barney's path created parallelograms, let's see if there is any relationship among them and the triangle ABC.


The perimeter of the parallelograms, excluding the segments that are part of the triangle, are equivalent to the perimeter of the triangle. This illustrates that Barney returns to the starting points on segment BC because his bouncing path is the same measure of the triangle.
What happens if Barney's starting point is point B or C?
If he begins at the points B or C, he will also return to his original resting place. Click here to see a diagram.
What happens if Barney's starting point outside to the triangle?
If Barney's starts at a point P outside to the triangle, not an element of lines AC, AB, or BC, then Barney will never return to his original position. Click here to see a diagram.
What do you think will happen starts bouncing from a point in the interior of the triangle, such as the centroid? Give it a try.