Bouncing Barney

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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?

First, find a point on the segment and construct a parallel line from the original point to a point on another line.

Follow Barney as he travels around the triangle.

The finding is that Barney will cross each segment twice.